Irregular Pairs Research Articles

Model theory and agnostic online learning via excellent sets

We use algorithmic methods from online learning to explore some important objects at the intersection of model theory and combinatorics, and find natural ways that algorithmic methods can detect and explain (and improve our understanding of) stable structure in the sense of model theory. The main theorem deals with existence of ϵ \epsilon -excellent sets (which are key to the Stable Regularity Lemma, a theorem characterizing the appearance of irregular pairs in Szemerédi’s celebrated Regularity Lemma). We prove that ϵ \epsilon -excellent sets exist for any ϵ > 1 2 \epsilon > \frac {1}{2} in k k -edge stable graphs in the sense of model theory (equivalently, Littlestone classes); earlier proofs had given this only for ϵ > 1 / 2 2 k \epsilon > 1/{2^{2^k}} or so. We give two proofs: the first uses regret bounds from online learning, the second uses Boolean closure properties of Littlestone classes and sampling. We also give a version of the dynamic Sauer-Shelah-Perles lemma appropriate to this setting, related to definability of types. We conclude by characterizing stable/Littlestone classes as those supporting a certain abstract notion of majority: the proof shows that the two distinct, natural notions of majority, arising from measure and from dimension, densely often coincide.

A Survey of Galaxy Pairs in the SDSS Photometric Images based on Faster-RCNN

Galaxy pairs hold significant importance in understanding the evolution of galaxies, and the extensive search for a large sample of galaxy pairs is meaningful. In this article, we develop a deep learning-based approach for the search of galaxy pairs and conduct a comprehensive search on Sloan Digital Sky Survey (SDSS) images. In nine million photometric images, 17,965 physical galaxy pairs with spectral or photometric redshifts are detected. Four sets of results are provided, including physical pairs determined by two spectral redshifts, two photometric redshifts, one spectral redshift, and one photometric redshift, and visual irregular pairs that have no precise redshift information but can be inferred as physical galaxy pairs based on the morphological changes. Then their morphological and physical characteristics are explored, the redshifts of most targets are around 0.1, and as the redshift difference between two galaxies increases, the number of galaxy pairs gradually reduces. The distributions of star formation rate (SFR) are not the same for different morphologies of galaxy pairs, irregular pairs have higher SFR than the other three types, and statistics indicate that the SFR of galaxies depends on both nearby galaxies and internal properties. Color and stellar mass are also key properties of galaxies which can reflect the status of galaxy pairs. Compared to other surveys, a greater number of galaxy pair targets are detected, and this is also the first extensive detection of galaxy pairs in SDSS images using photometric redshifts. These galaxy pair samples can greatly aid in the study of galaxy evolution.

Evidence of Absence: Abstract Metrical Structure in Speech Planning.

Rhythmic structure in speech is characterized by sequences of stressed and unstressed syllables. A large body of literature suggests that speakers of English attempt to achieve rhythmic harmony by evenly distributing stressed syllables throughout prosodic phrases. The question remains as to how speakers plan metrical structure during speech production and whether it is planned independently of phonemes. To examine this, we designed a tongue twister task consisting of disyllabic word pairs with overlapping phonological segments and either matching or non-matching metrical structure. Results showed that speakers had more difficulty producing metrically regular word pairs, compared to irregular pairs; that is, word pairs with irregular meter had faster productions and fewer speech errors in this production task. This finding of metrical regularity inhibiting production is inconsistent with an abstract metrical structure that is planned independently of phonemes at the point of phonological encoding.

The model structure of the hammerhead ribozyme formed by RNAs of reciprocal chirality.

RNA-based tools are frequently used to modulate gene expression in living cells. However, the stability and effectiveness of such RNA-based tools is limited by cellular nuclease activity. One way to increase RNA’s resistance to nucleases is to replace its D-ribose backbone with L-ribose isomers. This modification changes chirality of an entire RNA molecule to L-form giving it more chance of survival when introduced into cells. Recently, we have described the activity of left-handed hammerhead ribozyme (L-Rz, L-HH) that can specifically hydrolyse RNA with the opposite chirality at a predetermined location. To understand the structural background of the RNA specific cleavage in a heterochiral complex, we used circular dichroism (CD) and nuclear magnetic resonance (NMR) spectroscopy as well as performed molecular modelling and dynamics simulations of homo- and heterochiral RNA complexes. The active ribozyme-target heterochiral complex showed a mixed chirality as well as low field imino proton NMR signals. We modelled the 3D structures of the oligoribonucleotides with their ribozyme counterparts of reciprocal chirality. L- or D-ribozyme formed a stable, homochiral helix 2, and two short double heterochiral helixes 1 and 3 of D- or L-RNA strand thorough irregular Watson–Crick base pairs. The formation of the heterochiral complexes is supported by the result of simulation molecular dynamics. These new observations suggest that L-catalytic nucleic acids can be used as tools in translational biology and diagnostics.

Independent Control of the Chirality and Polarity for the Magnetic Vortex in Symmetric Nanodot Pairs

We studied the vortex states during the magnetization process for nanomagnetic dot pairs in different geometries, including a series of dual regular polygons with 4–16 sides and irregular shape dot pairs. All geometries demonstrated independent control of the vortex chirality and polarity and could be accomplished by adjusting the in-plane magnetic field direction. To achieve chirality and polarity control, both shape anisotropy and coupling interaction play a vital role. For the regular polygons, the effect of shape anisotropy wanes as the number of side increases, and the coupled interaction is enhanced relatively. According to the results, and combined with those for dual-circle and dual-rectangular magnetic disks, we state the principle for the geometry of the disk to achieve independent control of the chirality and polarity.

Preliminary Characterization of the Probiotic Properties of a Bacterial Strain for Used in Monogastric Nutrition

This study aimed to evaluate some probiotic properties of Bacillus subtilis ATCC 6051a. The phenotypic profile, resistance to pH by simulated gastric juice (pH 2 and 3), bile salts by simulated intestinal fluid, survivability (%), heat and antibiotics tolerance were investigated. The strain is a Gram-positive, rod-shaped bacteria, arranged in short chains or in small irregular pairs with the ability to produce spores. Good viability at pH 2 and 3, with a survival of more than ≥80%, was found. In the presence of bile salts 0.3%, over 4 h, the strain exhibited a survival ≥85%. At 80°C, for 120 min., the strain showed good growth (9.04 log CFU/ml). Results were sensitive to most antibiotics, with a highly susceptible (between 16 – 25 mm) to erythromycin, clindamycin, amoxicillin, chloramphenicol, ciprofloxacin, amikacin and kanamycin. The strain was found to be sensitive to vancomycin, gentamicin, and tetracycline. The present research demonstrated that Bacillus subtilis ATCC 6051a can survive under gastrointestinal conditions, which involves them to future in vitro and in vivo probiotic studies.

FMRI evidence for the involvement of the procedural memory system in morphological processing of a second language.

Behavioural evidence suggests that English regular past tense forms are automatically decomposed into their stem and affix (played = play+ed) based on an implicit linguistic rule, which does not apply to the idiosyncratically formed irregular forms (kept). Additionally, regular, but not irregular inflections, are thought to be processed through the procedural memory system (left inferior frontal gyrus, basal ganglia, cerebellum). It has been suggested that this distinction does not to apply to second language (L2) learners of English; however, this has not been tested at the brain level. This fMRI study used a masked-priming task with regular and irregular prime-target pairs (played-play/kept-keep) to investigate morphological processing in native and highly proficient late L2 English speakers. No between-groups differences were revealed. Compared to irregular pairs, regular pairs activated the pars opercularis, bilateral caudate nucleus and the right cerebellum, which are part of the procedural memory network and have been connected with the processing of morphologically complex forms. Our study is the first to provide evidence for native-like involvement of the procedural memory system in processing of regular past tense by late L2 learners of English.

Regularity lemmas for stable graphs

We develop a framework in which Szemeredi's celebrated Regularity Lemma for graphs interacts with core model-theoretic ideas and techniques. Our work relies on a coincidence of ideas from model theory and graph theory: arbitrarily large half-graphs coincide with model-theoretic instability, so in their absence, structure theorems and technology from stability theory apply. In one direction, we address a problem from the classical Szemeredi theory. It was known that the pairs in the statement of Szemeredi's regularity lemma cannot be eliminated, due to the counterexample of half-graphs (i.e., the order property, corresponding to model-theoretic instability). We show that half-graphs are the only essential difficulty, by giving a much stronger version of Szemeredi's regularity lemma for models of stable theories of graphs (i.e. graphs with the non-k�-order property), in which there are no irregular pairs, the bounds are significantly improved, and each component satisfies an condition. In the other direction, we take a more model-theoretic approach, and give several new Szemeredi-type partition theorems for models of stable theories of graphs. The first theorem gives a partition of any such graph into indiscernible components, meaning here that each component is either a complete or an empty graph, whose interaction is strongly uniform. This relies on a finitary version of the classic model-theoretic fact that stable theories admit large sets of indiscernibles, by showing that in models of stable theories of graphs one can extract much larger indiscernible sets than expected by Ramsey's theorem. The second and third theorems allow for a much smaller number of components at the cost of weakening the indivisibility condition on the components. We also discuss some extensions to graphs without the independence property. All graphs are finite and all partitions are equitable, i.e. the sizes of the components differ by at most 1. In the last three theorems, the number of components depends on the size of the graph; in the first theorem quoted, this number is a function ofonly as in the usual Szemeredi regularity lemma.

Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck 2/log* k pairs of parts which are not $${\epsilon}$$ -regular, where $${c,\epsilon >0 }$$ are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemerédi’s regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter $${\epsilon}$$ may require as many as $${2^{\Omega}(\epsilon^{-2})}$$ parts. This is tight up to the implied constant and solves a problem studied by Lovász and Szegedy.

Robust LDPC decoding using irregular decoders

It has been observed that irregular decoders for low density parity-check (LDPC) codes can be more robust to channel estimation errors compared to conventional decoders. In this work, by presenting a robustness measure, we propose a method for the joint optimization of irregular LDPC code-decoder pairs to have the widest convergence region when channel estimation errors exist.

On irregular prime power divisors of the Bernoulli numbers

Let $B_n$ ($n = 0, 1, 2, ...$) denote the usual $n$-th Bernoulli number. Let $l$ be a positive even integer where $l=12$ or $l \geq 16$. It is well known that the numerator of the reduced quotient $|B_l/l|$ is a product of powers of irregular primes. Let $(p,l)$ be an irregular pair with $B_l/l \not\equiv B_{l+p-1}/(l+p-1) \modp{p^2}$. We show that for every $r \geq 1$ the congruence $B_{m_r}/m_r \equiv 0 \modp{p^r}$ has a unique solution $m_r$ where $m_r \equiv l \modp{p-1}$ and $l \leq m_r < (p-1)p^{r-1}$. The sequence $(m_r)_{r \geq 1}$ defines a $p$-adic integer $\chi_{(p, l)}$ which is a zero of a certain $p$-adic zeta function $\zeta_{p, l}$ originally defined by T. Kubota and H. W. Leopoldt. We show some properties of these functions and give some applications. Subsequently we give several computations of the (truncated) $p$-adic expansion of $\chi_{(p, l)}$ for irregular pairs $(p,l)$ with $p$ below 1000.

Bernoulli numbers, Wolstenholme's theorem, and [formula omitted] variations of Lucas' theorem

In this note we shall improve some congruences of G.S. Kazandzidis and D.F. Bailey to higher prime power moduli, by studying the relation between irregular pairs of the form ( p , p − 3 ) and a refined version of Wolstenholme's theorem.

Irregular Primes to One Million

Using "fast" algorithms for power series inversion (based on the fast Fourier transform and multisectioning of power series), we have calculated all irregular primes up to one million, including their indices of irregularity and associated irregular pairs. Using this data, we verified that Fermat’s "Last Theorem" and Vandiver’s conjecture are true for these primes. Two primes with index of irregularity five were already known; we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six.

Irregular primes to one million

Using "fast" algorithms for power series inversion (based on the fast Fourier transform and multisectioning of power series), we have calculated all irregular primes up to one million, including their indices of irregularity and associated irregular pairs. Using this data, we verified that Fermat’s "Last Theorem" and Vandiver’s conjecture are true for these primes. Two primes with index of irregularity five were already known; we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six.

Supplement to: ``The Criteria of Kummer and Mirimanoff Extended to Include 22 Consecutive Irregular Pairs''

Supplement to: ``The Criteria of Kummer and Mirimanoff Extended to Include 22 Consecutive Irregular Pairs''

The Criteria of Kummer and Mirimanoff Extended to Include 22 Consecutive Irregular Pairs

The Criteria of Kummer and Mirimanoff Extended to Include 22 Consecutive Irregular Pairs

On Fermat's last theorem and the Bernoulli numbers

Using Kummer's criteria we show that if the first case of Fermat's last theorem fails for the prime p, then there exist irregular pairs satisfying certain relations.

Paired-Associate Analogue of Language Learning: Role of Reinforcement, Repetition, and Number of Rule Instances

Adults learned 25 paired associates, forming a 5 by 5 matrix, in which the stimuli were 2-digit numbers and the responses were 2-letter pairs; 23 pairs (“regulars”) followed matrix rules; 2 pairs (“irregulars”) did not. on study trials, Ss were presented 21 regular pairs once and 2 irregular pairs twice; test trials were with all 25 stimuli. Ss learned the pairs more rapidly than those in a similar study which used fewer pairs and a smaller matrix. Ss receiving verbal reinforcement for correct responses on test trials performed no better than Ss not reinforced. Ss receiving both verbal reinforcement and repetition of correct responses on test trials learned fastest. Performance on irregular pairs was initially better than on regular pairs. Errors on the irregulars involved generalization of regular responses.

On the learning of morphological rules: An experimental analogy

Three experiments were conducted using modifications of a paradigm used by Esper (1925) . Using the study-test procedure, S s learned 16 paired associates in which the stimuli were 2-digit numbers and the responses were 2-letter pairs. Each single digit was associated with a letter to form a four-by-four matrix of 2-digit 2-letter stimulus-response pairs. Sixteen college S s were presented 12 of the 16 pairs (Exp. I); 12 of the 16 pairs plus 4 irregular pairs (Exp. II); or 12 of the 16 pairs plus 2 irregular pairs (Exp. III), and were tested with the 16 stimuli in all cases. The irregular pairs were presented three times as often (Exp. II) or twice as often (Exp. III) as the regular pairs. It was observed that the omitted pairs were learned quickly after the regularities of the presented pairs were learned; the irregular pairs were learned more rapidly than the regular pairs, and errors on the irregular pairs took the form of overgeneralization of the regular responses. It was assumed that the experimental conditions were analogous to the learning of verb inflection by children and the results were remarkably similar to the behavior of young children acquiring the morphological rules of verb inflection.

Wear and Tear as Factors in the Textual History of the Quarto Version of King Lear

N an earlier study' I presented a theory designed to explain the omission of certain passages from the quarto texts of six of Shakespeare's plays, King Lear, Othello, Hamlet, Troilus A g b and Cressida, Richard Ill, and 2 Henry IV. Having observed that these passages appear in irregular pairs separated by some 40, 50, or 6o lines, I suggested that, as the dramatists of the time averaged 40, 50, or 6o lines on a manuscript page, using both sides of the paper and leaving no margins at top or bottom, these passages might originally have been written back to back at the foot of single leaves of the manuscript and then have disappeared as the lower edges of these leaves were worn away.2 The purpose of the present article is to test this theory by exploring its implications for the textual history of the quarto version of one of these six plays, King Lear. If accepted, this theory will, first of all, confirm present-day opinion that a number of passages not found in Q3 were accidentally omitted from the authentic text rather than subsequently added by Shakespeare.4 At the same time it could be assumed that isolated passages were worn away by attrition affecting one side of the leaf only. In the second place, the theory will offer a new explanation for many of that multitude of single-word variants between Q and F that characterizes this particular play. In a manuscript so badly worn as to lose the lower part of nine of its leaves, single words might have disappeared elsewhere-some partially, others completely-and then have been restored incorrectly by the compositor of the first quarto. If, for instance, the word disasters (I. i. i74), which is the F