A version numbering scheme with a useful lexicographical order

By the problem, called a representation of the integers, is meant the encoding of the positive integers without a known upper bound, for which unique decoding can be provided. From the viewpoint of information theory, this problem has been discussed theoretically in its relation to the universality, asymptotic optimality, and computation theory. The result in those approaches has been applied to the data compression scheme for the type of data referring to the past as well as the URR and other floating-pointnumber representations with variable exponential length. This paper discusses representation of the integers problem from the viewpoint of the codeword-length function, which is the function to represent the length of the codeword. As a result, a general construction method is presented which has the desirable property and can uniquely encode all representations. Several methods are given from this approach, which can construct not only the traditional representations of the integers but also the first representations of the integers which has not been considered. It is noted first that two construction methods of codeword length, which are independent of the encoding, are considered. One is based on the combination of the codeword-length functions, and the other is based on the codeword-length distribution. More precisely, it is shown that not only are the traditional representations derived, but also the representations that have not been considered in the past. Using detailed examples, it is shown that the computational complexity of the encoding and decoding depends on the codeword-length function.

In this paper we give a formula for the number of representations of some square-free integers by certain ternary quadratic forms and estimate the lower bound of the 2-power appearing in this number.

The topic of generating all subsets of a given set occupies an important place in the books on combinatorial algorithms. The considered algorithms, like many other generating algorithms, are of two main types: for generating in lexicographic order or in Gray code order. Both use binary representation of integers (i.e., binary vectors) as characteristic vectors of the subsets.Ordering the vectors of the Boolean cube according to their weights is applied in solving some problems—for example, in computing the algebraic degree of Boolean functions, which is an important cryptographic parameter. Among the numerous orderings of Boolean cubic vectors by their (Hamming) weights, two are most important. The weight ordering, where the second criterion for sorting the vectors of equal weights, is the so-called weight-lexicographic order. It is considered in detail in [2]. Considered here is the second weight ordering, where the vectors of equal weights are arranged so that every two consecutive vectors differ in exactly two coordinates, i.e., they are ordered by minimal change. The properties of this ordering are derived. Based on these, an algorithm was developed that generates the vectors of the Boolean cube in this ordering. It uses a binary representation of integers and only performs additions of integers instead of operations on binary vectors. Its time and space complexities are of a linear type with respect to the number of vectors generated. The algorithm was used in the creation of sequence A351939 in OEIS [8].Keywordsn-dimensional Boolean cubeBinary vectorWeight ordering relationMinimal changeGenerating all subsetsGray codeRevolving door algorithm

In this paper we propose very efficient itemset representation for frequent itemset mining from transactional databases. The combinatorial number system is used to uniquely represent frequent k-itemset with just one integer value, for any k ≥ 2. Experiments show that memory requirements can be reduced up to 300 %, especially for very low minimal support thresholds. Further, we exploit combinatorial number schema for representing candidate itemsets during iterative join-based approach. The novel algorithm maintains one-dimensional array rank, starting from k = 2nd iteration. At the index r of the array, the proposed algorithm stores unique integer representation of the r-th candidate in lexicographic order. The rank array provides joining of two candidate k-itemsets to be O(1) instead of O(k) operation. Additionally, the rank array provides faster determination which candidates are contained in the given transaction during the support count and test phase. Finally, we believe that itemset ranking by combinatorial number system can be effectively integrated into pattern-growth algorithms, that are state-of-the-art in frequent itemset mining, and additionally improve their performances.

A generalization of a combinatorial theorem of macaulay

The Riesz space E is said to be Archimedean if inf (n-1u: n = 1, 2,…) = 0 holds for every u ∈ E + .There exist non-Archimedean Riesz spaces. As an example, let E= ℝ2 with the lexicographical ordering (see Example 4.2(ii)). The element (0,1) in E is a lower bound of the sequence (n-1, n-1): n = 1, 2,…. Hence, u = (1,1) does not satisfy the condition that inf(n-lu: n = 1, 2,…) = 0. Actually, the sequence of all n-1u does not have an infimum at all in this case. Before presenting more examples, we prove a simple theorem.

Motivated by work of Erdős, Milner and Rado, we investigate symmetric and asymmetric partition relations for linear orders without the axiom of choice. The relations state the existence of a subset in one of finitely many given order types that is homogeneous for a given colouring of the finite subsets of a fixed size of a linear order. We mainly study the linear orders 〈α2,<lex〉, where α is an infinite ordinal and <lex is the lexicographical order. We first obtain the consistency of several partition relations that are incompatible with the axiom of choice. For instance we derive partition relations for 〈ω2,<lex〉 from the property of Baire for all subsets of ω2 and show that the relation \(\langle ^{\kappa }{2}, <_{lex}\rangle \longrightarrow (\langle ^{\kappa }{2}, <_{lex}\rangle )^{2}_{2}\) is consistent for uncountable regular cardinals κ with κ<κ = κ. We then prove a series of negative partition relations with finite exponents for the linear orders 〈α2,<lex〉. We combine the positive and negative results to completely classify which of the partition relations \(\langle ^{\omega }{2}, <_{lex}\rangle \longrightarrow (\bigvee _{\nu <\lambda }K_{\nu },\bigvee _{\nu <\mu }M_{\nu })^{m}\) for linear orders Kν,Mν and m≤4 and 〈ω2,<lex〉→(K,M)n for linear orders K,M and natural numbers n are consistent.

of natural numbers feature prominently in almost a l l parts of mathematics, as well as many areas outside the t rad i t i ona l boundaries of the subject. Neil Sloane's of Integer Sequences [11] bears testimony to th is fact . I t is a l i s t of 2372 sequences, in lex icographicorder, drawn from a wide range of topics. The only c r i t e r i a for inc lus ion of a sequence of natural numbers in the Handbook are that enough terms should be known to d is t ingu ish i t from i t s neighbours, and that somebody must have found i t in terest ing enough to commit i t to p r i n t in the s c i e n t i f i c l i t e r a ture.

Representations of integers by linear forms in nonnegative integers

It is clear that, for k ≥ 4, every nonnegative integer is representable as a sum of k squares. Indeed, one can always write \( n = \sum\nolimits_{{i = 1}}^4 {x_i^2} + {0^2} + \cdot \cdot \cdot + {0^2} \), with an arbitrary number of zeros. On the other hand, the number of representations r k (n) increases very rapidly with n (see Chapter 12), and besides the representations with k — 4 zeros, one usually finds others with fewer zeros or none at all. For example, if k = 5, then 5 = 22 + 12 + 02 + 02 + 02 = 12 + 12 + 12 + 12 + 12.

Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with n≥2, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent to r modulo n. Similar expressions are also dealt with for the case of Lucas numbers. Symmetric and anti-symmetric properties of Fibonacci and Lucas numbers are used in the proofs.

Positive definite binary hermitian forms with finitely many exceptions

In the paper entitled ‘Asymptotic formula in a generalized Waring's problem’, I established an asymptotic formula for the number of representations of a large natural number N in the formwhere x1, x2, …, x7 and k are natural numbers with k ≥ 4 (see (2) Theorem 2).

For a matrix M and a positive integer r, the rank r rigidity of M is the smallest number of entries of M which one must change to make its rank at most r. There are many known applications of rigidity lower bounds to a variety of areas in complexity theory, but fewer known applications of rigidity upper bounds. In this paper, we use rigidity upper bounds to prove new upper bounds in a few different models of computation. Our results include: - For any d>1, and over any field F, the N × N Walsh-Hadamard transform has a depth-d linear circuit of size O(d · N1 + 0.96/d). This circumvents a known lower bound of Ω(d · N1 + 1/d) for circuits with bounded coefficients over ℂ by Pudlak (2000), by using coefficients of magnitude polynomial in N. Our construction also generalizes to linear transformations given by a Kronecker power of any fixed 2 × 2 matrix. - The N × N Walsh-Hadamard transform has a linear circuit of size ≤ (1.81 + o(1)) N log2 N, improving on the bound of ≈ 1.88 N log2 N which one obtains from the standard fast Walsh-Hadamard transform. - A new rigidity upper bound, showing that the following classes of matrices are not rigid enough to prove circuit lower bounds using Valiant’s approach: (1) for any field F and any function f : {0,1}n → F, the matrix Vf ∈ F2n × 2n given by, for any x,y ∈ {0,1}n, Vf[x,y] = f(x ∧ y), and (2) for any field F and any fixed-size matrices M1, …, Mn ∈ Fq × q, the Kronecker product M1 ⊗ M2 ⊗ ⋯ ⊗ Mn. This generalizes recent results on non-rigidity, using a simpler approach which avoids needing the polynomial method. - New connections between recursive linear transformations like Fourier and Walsh-Hadamard transforms, and circuits for matrix multiplication.

Even among RNA viruses, which generally exhibit high evolutionary plasticity due to low fidelity of their RNA polymerases, HIV-1 is second only to HCV for its ability to generate within-host genetic diversity [1]. HIV's rapid generation time leads to this high genetic diversity. The unfortunate consequences of HIV's rapid evolution are resistance to antiretroviral drugs [1], partial escape from immune responses [2]–[4], the ability to switch tropism for target cells [5], and potential threats to new therapeutic strategies [6], [7]. The forces driving and influencing HIV evolution include Darwinian selection, limited population size, linkage, recombination, epistasis, spatial aspects, and dynamic factors (particularly due to the immune response). These factors, and the parameters that define them, can be difficult to discern. One of the most elusive parameters critically important for the rate of evolution in every medically relevant scenario is the “effective population number” (Ne ff) (Figure 1). By definition, the census population size of HIV is the total number of infectious proviruses integrated into the cellular DNA of an individual at a given time. However, the genetically relevant Ne ff may differ substantially from the census population size. In this volume of PLOS Genetics, Pennings and colleagues [8] use new insights into “hard” and “soft” selective sweeps to estimate the effective population size of HIV. Figure 1 Beneficial viral mutants (red) arise in the “effective” virus subpopulation (N eff, pink circle) and spread gradually to the entire “census” population (blue circle). The search for N eff (and other HIV evolutionary parameters) has gone on for almost two decades, following every turn and hitting each pothole on the eventful road of HIV modeling [9]. The rapidity of resistance to monotherapy (in 1–2 weeks) was explained by the deterministic selection of alleles that preexist therapy in minute quantities [1]. The large numbers of virus-producing cells (∼108) in the lymphoid tissue of experimentally infected macaques seemed to confirm this simple Darwinian selection model [10]. However, the Darwinian view has faced challenges. Tajima's “neutrality test” applied to HIV sequences in untreated patients assumed that selection was neutral and predicted much smaller “effective” populations, of N eff∼103 [11]. Since Tajima's approach was designed to detect isolated selective sweeps at one or a few mutant sites—while HIV exhibits hundreds of diverse sites in vivo—two groups re-tested the result. A linkage disequilibrium (LD) test [12] and analysis of the variation in the time to drug resistance [13] arrived at the same value, N eff = (5–10)×105, for an average patient (with the mutation rate ∼10−5 per base). Such populations are sufficiently large for deterministic selection to dominate, yet not large enough to neglect stochastic effects altogether. The LD test [12] is affected by recombination, and HIV's recombination rate had not been well measured at that time. The recent measurement of 5×10−6 crossovers per base per HIV replication cycle in an average untreated individual [14]–[16] updates N eff to (1–2)×105, not far from the original value. A recent study of the pattern of diversity accumulation in early and late HIV infection confirms the range of N eff [17]. However, all these estimates of N eff are lower bounds. Pennings et al. [8] continue this quest for an effective population size of HIV using a new method based on a theoretical calculation of the probability of multiple introductions of a beneficial allele at a site before it is fixed in a population [18]. The prediction does not depend on whether mutations are new or result from standing variation prior to therapy. The authors use sequence data obtained from 30 patients who failed suboptimal antiretroviral regimens, including efavirenz [19]—a non-nucleoside reverse transcriptase (RT) inhibitor (NNRTI)—and who exhibited a rise of drug-resistant alleles in RT. The sequence data reveal fixation of two alleles, both corresponding to an amino-acid replacement K103N. Pennings et al.'s analysis focuses on the genetic composition at RT codon 103 and the adjacent 500 nucleotides. Based on the changes in the genetic diversity in this region, 30 fixations are classified into “hard” selective sweeps with a single parental sequence, or “soft” sweeps with multiple parental sequences. Observing that both types of sweep occurred at similar frequencies (also confirmed by observations in other resistance codons), the authors predict N eff = 1.5×105, in agreement with the LD test. Pennings et al. also discuss why “selectively neutral” methods based on synonymous diversity underestimate the population size. It is well known that a selection sweep lowers the diversity at linked sites (hence the term “sweep”) and any method assuming selective neutrality translates lower diversity to smaller N eff. The interesting part is the dynamic component of this effect. Pennings et al. demonstrate that rapid sweeps are followed by long periods when the diversity recovers at the linked sites (for synonymous sites, these periods are very long). From another angle, we can add that selection shortens the time to the common ancestor, which decreases the sequence divergence. The ancestral-tree argument is rather general and also applies to a large number of linked sites evolving under selection [20]–[23]. The previous estimates [12], [13], [17] were lower bounds on N eff. In contrast, the Pennings et al. study puts a number on N eff. However, this number (N eff = 1.5×105) raises a question: why is N eff so far below the census population size of 108 or more? Pennings et al. offer an elegant explanation of this relatively small N eff in the spirit of the “traveling wave” approach [24]–[27]. They note that resistant alleles at different sites emerge against different fitness backgrounds. To be fixed, alleles conferring a small benefit must emerge in the most-fit genomes [28], [29]; hence, the effective N eff for these alleles is small. Alleles with a larger beneficial effect can explore a larger fraction of population (larger N eff). Conceptually, this idea is quite correct; quantitatively, in the context of drug resistance, some problems arise. For example, the fitness benefit from a resistance mutation (under drug) is almost 100%, while the difference between the fittest and the average genome (in untreated patients) is a modest ∼10% [14]. Indeed, the average selection coefficient is quite small, ∼0.5% [14], [15]. There may be several other reasons for N eff<108, as follows. By considering only 500 bases (∼5%) of the HIV genome, the study may underestimate the number of genetic backgrounds in which the resistant allele can be observed. N eff is likely to vary in time—similar to viremia, which decays strongly after the onset of therapy and rebounds after its failure—and the placement of the inferred population size within the therapy time frame is unclear. Specifically, it is unclear from the empirical source [19] whether K103N mutations are generated before therapy (which is likely, considering that the mutation of interest decays very slowly in vivo in untreated patients and therefore has a low mutation cost [30]) or after therapy fails for another reason (see Figure 1 in [19]). In the first scenario, inferred N eff = 105 is the pretreatment number. In the second scenario, the pretreatment number must be much higher than 105, since the replicating census population is reduced by a large factor (∼100) following initiation of therapy. Other factors, such as variation of the population number among patients and the spatial organization of the infected tissue [31] (both neglected in the test), may be relevant. Furthermore, the authors' calculations rely on the assumption of equal mutation rates for the two resistance mutations analyzed (both transversions). If the underlying rate of AAA to AAC is much greater than that of to AAT, the cited analysis would have underestimated the frequency of soft sweeps, yielding an underestimate of N eff. A significant complicating factor is the presence, in the parent study [19], of other drugs, particularly the nucleoside RT inhibitors (NRTIs) AZT and 3TC. In some cases, mutations conferring resistance to these drugs may have also contributed to failure (e.g., during the precursor monotherapy; see Figure 1 in [19]), and the requirement for these additional changes would have made the frequency of resistant strains much less than the estimate. For virus that escaped the combination treatment in the absence of NRTI mutations, replication was most likely occurring only in a fraction, or “sanctuary,” of cells that did not receive an inhibitory dose of these drugs. Either or both of these effects would have led to a potentially large underestimate of N eff. Indeed, a recent study of rapid NNRTI resistance, in SIV-infected monkeys treated with efavirenz monotherapy, used an ultrasensitive PCR assay to estimate the pre-therapy level of either K103N mutation as less than 0.0001% [32], implying a total replicating population of >106. For these reasons, the value N eff = 1.5×105 obtained in the study of Pennings et al. should probably still be regarded as a lower bound. At the same time, the study solidifies our understanding of HIV evolution as a Darwinian process and leads to important questions regarding the structure of HIV population, which are still waiting for new insights.