Bioinformatics is a rapidly developing field enabling scientific experiments through computer models and simulations. Considering the vast databases of biological data available, it is ex-tremely important to develop efficient methods and algorithms for their processing. Sequence comparison is the best method for studying the evolutionary interaction between genes. It is based on alignment – the process of arranging two or more sequences to achieve the maximum level of identity and degree of similarity. The paper presents a new version of the algorithm for pairwise DNA sequences alignment, based on a new method called CAT, where a dependency with a previous match and the closes neighbor are taken in consideration to increase uniqueness of the CAT profile and to reduce possible collisions, i.e. two or more sequence having same CAT profiles. This makes proposed algorithm suitable for finding exact match of a concrete DNA se-quence in a big set of DNA data faster. The generation of CAT profiles is made once before data has been uploaded to the database, allowing the profiles to be used as metadata for the sequenc-es. It consists of an algorithm to calculate a CAT profile against the selected reference sequences and an algorithm to compare two sequences based on the calculated CAT profiles. Improve-ments in generation of the CAT profiles, are detailed described in the paper. Block scheme, pseudo code tables and figures were updated according to the proposed new version and ex-perimental results. Experiments have been carried out with the new version of the method and different datasets to align DNA sequences based on the CAT method. New experimental results in terms of collisions, speed, and efficiency of the proposed solutions are presented. Experiments related to the performance comparison with Needleman-Wunsch were re-executed with the new version of the algorithm to confirm that we have the same performance. An analysis of the per-formance of the proposed CAT based algorithm against Knuth–Morris–Pratt algorithm, which has a complexity of O(n) and is one of the most widely used for searching biological data, was performed. The impact of prior matching dependencies on uniqueness for generated CAT pro-files is investigated. The analysis of the experimental results obtained by sequence alignment shows a small deviation of the proposed algorithm based on the CAT method, which can be ig-nored if this deviation is acceptable at the expense of performance. The time efficiency of the CAT algorithm remains constant, regardless of the length of the sequences. Therefore, the ad-vantage of the proposed method is its fast processing in the alignment of large sequences, for which the execution of the exact algorithms takes a long time. The example code realization of the CAT Method, under the protection of the GNU General Public License v3.0, can be accessed on GitHub at: https://github.com/HristoS/CATSequenceAnalysis.

In this paper, we introduce and study the notion of rough I₂-lacunary statistical convergence of double sequences in normed linear spaces. We also introduce the notion of rough I₂-lacunary statistical limit set of a double sequence and discuss about some properties of this set.

In the process of transcription initiation by RNA polymerase, promoter DNA sequences affect multiple reaction pathways determining the productivity of transcription. However, the question of how the molecular mechanism of transcription initiation depends on sequence properties of promoter DNA remains poorly understood. Here, combining the statistical mechanical approach with high-throughput sequencing results, we characterize abortive transcription and pausing during transcription initiation by Escherichia coli RNA polymerase at a genome-wide level. Our results suggest that initially transcribed sequences enriched with thymine bases represent the signal inducing abortive transcription. On the other hand, certain repetitive sequence elements broadly embedded in promoter regions constitute the signal inducing pausing. Both signals decrease the productivity of transcription initiation. Based on solution NMR and in vitro transcription measurements, we also suggest that repetitive sequence elements of promoter DNA modulate the rigidity of its double-stranded form, which profoundly influences the reaction coordinates of the productive initiation via pausing.

Despite its remote discovery (1977) and numerous reports of its presence in vari-ous plant species in many countries, the available molecular information for the artichoke Italian latent virus (AILV) is still limited to a single complete genome sequence (RNA1 and 2) isolate from grapevine (AILV-V) and to a partial sequence portion from RNA2 of an isolate of unknow origin and host, both reported in the GenBank. Here we report the results of molecular analyses conducted on RNA2 of AILV isolates, sequenced for the first time in this study, together with the first-time finding of AILV in a new host plant species, i.e., chard (Beta vulgaris subsp. vulgaris). The different AILV isolates sequenced are from artichoke (AILV-C), gladiolus (AILV-G), sonchus (AILV-S) and chard (AILV-B). At the bio-logical level, the AILV-B was found to be involved with vein clearing and mott-ling symptoms on chard leaves. At the molecular level, sequencing results of RNA2 segments showed that AILV-C, AILV-G, AILV-S and AILV-B were 4,629 nt in length (excluding the 3’ terminal polyA tail), hence one nt shorter than that of AILV-V reported in the GenBank. The comparative sequence analysis between the CR of RNA2 of all isolates showed that AILV-V was the most variable isolate, with the lowest sequence identities of 83.2% and 84.7% at the nucleotides and amino acids levels, respectively. Putative intra-species sequence recombination sites were predicted among the AILV isolates, of which the AILV-V, AILV-C and AILV-B genomes were found to be mostly involved. This study adds further mo-lecular information on the previously unrecorded high genomic variability of the Nepoviruses of subgroup B, together with information on recombinations that may have conditioned the infection and transmission of AILV in nature; thus learning the way forward to design a new integrated pest management of chard plantation.

Viruses are abundant and diverse entities that have important roles in public health, ecology, and agriculture. Identification and surveillance of viruses rely on understanding their genome organization, sequences, and replication strategy. Despite technological advancements in sequencing methods, our current understanding of virus diversity remains incomplete, highlighting the need to explore undiscovered viruses. Virus databases play a crucial role in providing access to sequences, annotations and other metadata, and analysis tools for studying viruses. However, there has not been a comprehensive review of virus databases in the last five years. This study aimed to fill this gap by identifying 24 active virus databases and included an extensive evaluation of their content, functionality and compliance with the FAIR principles. In this study, we thoroughly assessed the search capabilities of five database catalogs, which serve as comprehensive repositories housing a diverse array of databases and offering essential metadata. Moreover, we conducted a comprehensive review of different types of errors, encompassing taxonomy, names, missing information, sequences, sequence orientation, and chimeric sequences, with the intention of empowering users to effectively tackle these challenges. We expect this review to aid users in selecting suitable virus databases and other resources, and to help databases in error management and improve their adherence to the FAIR principles. The databases listed here represent current knowledge of viruses and will help aid users find databases of interest based on content, functionality, and scope. Use of virus databases is integral to gain new insights into the biology, evolution, and transmission of viruses, and develop new strategies to manage virus outbreaks and preserve global health.

The purpose of this study is to provide an accurate formula for calculating entropy for short DNA sequences and to demonstrate how to use it to examine leukemia patient surviving. We used IDIBAPS leukemia patient’s data base with 117 anonymized records. The generalized form of the Robust Entropy Estimator (EnRE) for short DNA sequences was proposed and key EnRE futures was showed. The Survival Analysis has been done using statistical package IBM SPSS. Entropy EnRE were calculated for leukemia patients for two samples: A. 2 groups divided by median EnRE and B. 2 groups of patients were formed according to their belonging to 1st and 4th quartiles of EnRE. The result of survival analysis are statistically significant: A. p < 0.05; B. p < 0.005. The death hazard for a patient with EnRE below median is 1.556 times that of a patient with EnRE over median and that the death hazard for a patient of 1st quartile (lowest EnRE) is 2.143 times that of a patient of 4th quartile (highest EnRE). The transition from median to quartile patients’ groups with more EnRE differentiation confirmed the unique significance of the entropy of DNA sequences for leukemia patients surviving.

In this paper, it is proved that s-iteration with error and Picard-Mann iteration with error iterative processes converge strongly to the unique fixed point of Lipschitzian strongly pseudo-contractive map. This convergence is almost -stable and -stable. Applications of these results to the operator equations, where is a strongly accretive and accretive mappings of into itself.

An experimental ultra wideband ground penetrating radar based on Golay complementary sequences is proposed to locate underground pipes. Golay complementary sequences with the code length of 1024 and frequency of 1 GHz are used as the probe signals. Two-dimensional image of the buried pipes is achieved by correlation method and back projection algorithm. The experimental results show that both the plastic pipe and metallic pipe can be located with a range resolution of 10 cm. Furthermore, as the Golay complementary sequences are a pair of complementary sequences, the sum of their correlation function yields twice the value of the peak at the target position and zero elsewhere. Thus, compared with the stepped frequency signal radar or chaotic signal radar, the Golay-based radar can significantly improve the signal noise ratio and has capability of deep detection.

The superfamily Lauxanioidea is a significant dipteran clade including over 2500 known species in three families: Lauxaniidae, Celyphidae and Chamaemyiidae. We sequenced the first five (three complete and two partial) lauxanioid mitochondrial (mt) genomes, and used them to reconstruct the phylogeny of this group. The lauxanioid mt genomes are typical of the Diptera, containing all 37 genes usually present in bilaterian animals. A total of three conserved intergenic sequences have been reported across the Cyclorrhapha. The inferred secondary structure of 22 tRNAs suggested five substitution patterns among the Cyclorrhapha. The control region in the Lauxanioidea has apparently evolved very fast, but four conserved structural elements were detected in all three complete mt genome sequences. Phylogenetic relationships based on the mt genome data were inferred by Maximum Likelihood and Bayesian methods. The traditional relationships between families within the Lauxanioidea, (Chamaemyiidae + (Lauxaniidae + Celyphidae)), was corroborated, however, the higher level relationships between cyclorrhaphan superfamilies are mostly poorly supported.

The world is facing a major health crisis, the global pandemic of COVID-19 caused by the SARS-CoV-2 coronavirus, for which no approved antiviral agents or vaccines are currently available. Here we describe a collection of codon-optimized coding sequences for SARS-CoV-2 cloned into Gateway-compatible entry vectors, which enable rapid transfer into a variety of expression and tagging vectors. The collection is freely available via Addgene. We hope that widespread availability of this SARS-CoV-2 resource will enable many subsequent molecular studies to better understand the viral life cycle and how to block it.

Examining the similarity of event environments or surroundings, more precisely settings, provides additional insight in analyzing event sequences as it provides information about the context and potential common factors that may have influenced them. This article proposes a new similarity measure for event setting sequences, which involve the space and time in which events occur. While similarity measures for spatiotemporal event sequences have been studied, the settings and setting sequences have not yet been studied. While modeling event setting sequences we consider spatial and temporal scales to define the bounds of the setting and incorporates dynamic variables alongside static variables. Using a matrix-based representation and an extended Jaccard index we developed new similarity measures that allow for the use of all variable data types. We successfully used these similarity measures coupled with other multivariate statistical analysis approaches in a case study involving setting sequences and pollution event sequences associated with the same monitoring stations, which validate the hypothesis that more similar spatial-temporal settings or setting sequences may generate more similar events or event sequences. In conclusion, these similarity measures have many potential real-world applications, and offer researchers a powerful tool for understanding different factors and their dynamics corresponding to occurrences of spatiotemporal event sequences.

Despite advancements in antiretroviral therapy (ART) that reduces the viral load to undetectable levels, viral eradication has not been achieved due to HIV-1 persistence in resting CD4+ T-cells. We, therefore, characterized the tat gene, which is essential for HIV-1 replication, from 20 viro-logically controlled HIV-infected (HIV+) older patients on long-term ART with improved CD4+ T-cell counts. PBMC genomic DNA from HIV+ patients were used to amplify tat gene by PCR fol-lowed by nucleotide sequencing and analysis. Phylogenetic analysis showed that each patient’s tat sequences were confined to their own subtrees and well discriminated from other patients’ sequences. Moreover, there was a low degree of viral heterogeneity and lower estimates of genetic diversity within these patients’ tat sequences. Most patients’ Tat deduced amino acid sequences showed intact open reading frames and maintained the important functional domains for Tat functions, including transactivation, TAR binding and nuclear localization. Furthermore, Tat deduced amino acid sequences showed variation in previously characterized cytotoxic T lym-phocytes (CTL) epitopes, suggesting escape mutants. In conclusion, a low degree of genetic varia-bility and conservation of functional domains and variations in CTL epitopes were the features of tat sequences from 20 HIV+ older patients with undetectable viral load on long-term ART.

With the outbreak of the COVID-19 disease, the research community is producing unprecedented efforts dedicated to better understand and mitigate the affects of the pandemic. In this context, we review the data integration efforts required for accessing and searching genome sequences and metadata of SARS-CoV2, the virus responsible for the COVID-19 disease, which have been deposited into the most important repositories of viral sequences. Organizations that were already present in the virus domain are now dedicating special interest to the emergence of COVID-19 pandemics, by emphasizing specific SARS-CoV2 data and services. At the same time, novel organizations and resources were born in this critical period to serve specifically the purposes of COVID-19 mitigation, while setting the research ground for contrasting possible future pandemics. Accessibility and integration of viral sequence data, possibly in conjunction with the human host genotype and clinical data, are paramount to better understand the COVID-19 disease and mitigate its effects.

Current knowledge of the RNA world indicates two different genetic codes being present throughout the living world. In contrast to non-coding RNAs that are built of repetitive nucleotide syntax, the sequences that serve as templates for proteins share – as main characteristics – a non-repetitive syntax. The differences in their syntax structure is coherent with the difference of the functions they represent. Whereas non-coding RNAs build groups that serve as regulatory tools in nearly all genetic processes, the coding sections represent the evolutionarily successful function of the genetic information storage medium. The DNA genomes themselves are rather inactive, whereas the non-coding RNA domain is highly active, even as non-random genetic innovation operators. This indicates that repetitive syntax is the essential pre-requisite for RNA interactions to install variable RNA-group-identities, whereas the non-repetitive syntax serves as a stable conservation tool for successful selection processes out of RNA-groups cooperation and competition. The interaction opportunities of RNA loops with repetitive syntax are higher than with non-repetitive ones. Interestingly, these two genetic codes resemble the function of all natural languages, i.e., (a) everyday language use for organization and coordination of biotic group behavior, and (b) artificial (instrumental) language use for conservation of blueprints for complex protein-body constructions.

The increasing availability of longitudinal data (repeated numerical observations of same units at different times) requires the development of flexible techniques to automatically detect common errors in such data. Besides obvious and easily identifiable cases, such as missing or out-of-range data, large longitudinal dataset often present problems not easily traceable by the techniques used for generic datasets. In particular, elusive and baffling problems are i) inversion of one or more values from one unit to another; ii) anomalous jumps in the series of values, iii) errors in the timing of the values due to a recalculation operated by the data providers to compensate previous errors. This work proposes a statistical-mathematical approach based on a system of indicators that is able to capture the complexity of the described problems by working at the formal level, regardless of the specific meaning of the data. The proposed approach identifies suspect erroneous data and is applicable in a variety of contexts. We implement this approach in a relevant database of European Higher Education institutions (ETER) by analyzing Total academic staff, that is one of the most important variables, used in empirical analysis as proxy of size and also considered by policy makers at European level.

In this study, we once again use a set of Fibonacci-like sequences to examine the symmetries within the genetic code. This time, our focus is on the physiological state of the amino acids, considering them as charged, in contrast to our previous work where they were seen as neutral. In a pH environment around 7.4, there are four charged amino acids. We utilize the properties of our sequences to accurately describe the symmetries in the genetic code table. These include Rumer’s symmetry, the third-base symmetry and the "ideal" symmetry, along with the "supersymmetry" classification schemes. We also explore the special chemical structure of the amino acid proline, presenting two perspectives—shCherbak’s view and the Downes-Richardson view, which perspectives are included in the description of the above-mentioned symmetries. Our investigation employs also elementary modular arithmetic to precisely describe the chemical structure of proline, connecting the two views seamlessly. Finally, our Fibonacci-like sequences prove instrumental in quickly establishing the multiplet structure of non-standard versions of the genetic code. We illustrate this with an example, showcasing the efficiency of our method in unraveling the complex relationships within the genetic code.

Collatz conjecture (3n+1 problem) is an application of Cantor's isomorphism theorem (Cantor-Bernstein) under recursion. The set of 3n+1 for all odd positive integers n, is an order isomorphism for (odd X, 3X+1). The other (odd X, 3X+1) linear order has been discovered as a bijective order-embedding, with values congruent to powers of four. This is demonstrated using a binomial series as a set rule, then showing the isomorphic structure, mapping, and cardinality of those sets. Collatz conjecture is representative of an order machine for congruence to powers of two. If an initial value is not congruent to a power of two, then the iterative program operates the (odd X, 3X+1) order isomorphism until an embedded value is attained. Since this value is a power of four, repeated division by two tends the sequence to one. Because this same process occurs, regardless of the initial choice for a positive integer, Collatz conjecture is true.

The meticulous study of finite automata has produced many important and useful results. Automata are simple yet efficient finite state machines that can be utilized in a plethora of situations. It comes, therefore, as no surprise that they have been used in classic game theory in order to model players and their actions. Game theory has recently been influenced by ideas from the field of quantum computation. As a result, quantum versions of classic games have already been introduced and studied. The PQ penny flip game is a famous quantum game introduced by Meyer in 1999. In this paper we investigate all possible finite games that can be played between the two players Q and Picard of the original PQ game. For this purpose we establish a rigorous connection between finite automata and the PQ game along with all its possible variations. Starting from the automaton that corresponds to the original game, we construct more elaborate automata for certain extensions of the game, before finally presenting a semiautomaton that captures the intrinsic behavior of all possible variants of the PQ game. What this means is that from the semiautomaton in question, by setting appropriate initial and accepting states, one can construct deterministic automata able to capture every possible finite game that can be played between the two players Q and Picard. Moreover, we introduce the new concepts of a winning automaton and complete automaton for either player.

Klebsiella pneumoniae is a leading cause of healthcare acquired infections. The severity of the infections is multifactorial including virulence and antimicrobial resistance properties. The objective of the study was to investigate the virulence determinants of K. pneumoniae strains colonizing patients hospitalized in the intensive care units (ICU) of a tertiary hospital in Uganda. A hospital based cross-sectional study involving 103 patients admitted in the ICU of Mulago National Referral hospital was conducted from January 2022 to April 2022. A total of 31 identified clinical strains of K. pneumoniae isolated from rectal swabs of the same number of patients were characterized. Susceptibility testing of colistin was performed by broth microdilution method, thereafter, 5 colistin resistant K. pneumoniae were purposefully selected for whole genome sequencing. Bioinformatics analysis was performed using established web-based tools to detect virulence genes, sequence type and other genomic features. Numerous virulence factor genes related to pathogenicity of K. pneumoniae were identified. They ranged from those related to colonization to those of persistence and dissemination including capsule genes (wzi), siderophore systems; yersiniabactin (ybt15), aerobactin (iutA), enterobactin (entC, entD, entF, entS, fepB, fepD, fepG, fes), and salmochelin (iroN), secretion systems; type II (T2SS) and type VI (T6SS-1). Furthermore, the isolates harbored outer membrane protein (ompA) gene, biofilm synthesis gene (fimH) and several adhesion genes. The results demonstrate the dual-risk of K. pneumoniae strains combining high level of virulence and colistin resistance which is a great concern as it leads to unmanageable infection outbreaks with high morbidity and mortality.

The Pearl River Delta (PRD) has undergone complex geological development within a multi-island faulted basin, shaped by the interplay of regional tectonic movements, Quaternary sea-level fluctuations, and fluvial-marine interactions. Despite a great number of studies on the Holocene sedimentary sequences and spatial difference of lithofacies and environments, scant attention has been paid to the overarching human influence on deltaic evolution and coastline modifications since the Neolithic epoch. To further elucidate the spatial variation in Holocene sedimentation and its underlying basement topography shaped during the Last Glacial Maximum (LGM), we compiled a comprehensive dataset incorporating borehole data from over 2,800 cores within the PRD. Subsequently, high-resolution isobath maps of Quaternary deltaic deposits were generated, offering unprecedented insights into sediment distribution. This dataset facilitated a nuanced reconstruction of pre-Holocene topography, revealing a zone characterized by elongated, deep-incised valleys governed by NW-SE fault orientations. Further, we delineated coastline shifts since the period of maximum marine transgression (~7000 years BP), contributing to an enhanced understanding of the formation and evolutionary patterns of the delta and the river network oscillations. Our findings illuminate increasing anthropogenic impact on the rate of fluvial sedimentation and land growth, particularly accentuated over the last two millennia, favoring deltaic accretion. The current study has considerable implications for understanding the dynamic interplay between land-sea interactions and anthropogenic activities in shaping the delta, and stands to significantly inform land and resource management strategies, as well as future urban planning within the delta's bay areas.

Astroviruses (AstVs) are occurs globally and are common causes of gastroenteritis in human and animals. The genetic diversity and epidemiology of AstVs in Africa is not well known, hence, we aimed to genetically characterize astroviruses in asymptomatic smallholder piglets in East Africa. Twenty-four samples randomly selected from 446 piglets (<6 months old), initially collected for rotavirus study, was sequenced for metagenomic analysis. Thirteen (13/24) samples had contigs with high identity to genus Mamastrovirus. Analysis of 7 strains with complete (or near complete) genome revealed variable nucleotide and amino acid sequence identities with known PoAstV strains. The U083 and K321 strains had nucleotide sequence similarities ranging from 66.4 to 75.4 % to the known PoAstV2 strains, nucleotide sequence similarity of U460 strain with known PoAstV3 ranged 57.0 to 65.1 % to the, while K062, K366, K451, and K456 strains showed nucleotide sequence similarities of 63.5 to 80 % to the known PoAstV4 strains. The low sequence identities (<90 %) indicate that novel genotypes of PoAstVs are circulating in the study area. Multiple recombination events were detected in our PoAstV4 strains, indicating that the genetic diversity observed in these strains may be due to recombination. Importantly, we identified potential candidate epitopes with conserved peptides in our PoAstV strains that could aid in the design of immune diagnosis tools and subunit vaccines. Our data provide new intuitions into the genetic structure of porcine astroviruses in East African.

Impressing discoveries in the field of the genetic code have been described by its researchers by means of the terminology borrowed from linguistics and the theory of communications. Leading experts on structural linguistics believe for a long time already that languages of human dialogue were formed not from an empty place, but they are continuation of genetic language or, anyhow, are closely connected with it, confirming the idea of information commonality of organisms. The aticle continues the theme about a connection of linquistic languages with the genetic language. It describes results of comparative study of long Russian literary texts (novels by L.Tolstoy, F.Dostoevsky, A.Pushkin, etc.) and long sequences of hydrogen bonds in double helixes of DNA of different organisms. Formalisms of quantum informatics are used in modeling some of these results taking into account thoughts of many researches about possible using principles of quantum informatics in organisation of living bodies.

The fractional Hilbert transform, a generalization of the Hilbert transform, has been extensively studied in the literature because of its extensive use in optics, engineering, and signal processing. In the present work, we aimed to expand the fractional Hilbert transform to a space of generalized functions known as Boehmians. We introduce a new fractional convolution operator for the fractional Hilbert transform to prove a convolution theorem similar to the classical Hilbert transform and also to extend the fractional Hilbert transform to Boehmians. We also construct a suitable Boehmian space on which the fractional Hilbert transform exists. Further, we investigate convergence of the fractional Hilbert transform for the class of Boehmians and discuss the continuity of the extended fractional Hilbert transform.

Two X chromosomes of female mammals randomly inactivate one of paternal or maternal X chromosome in early embryonic development and all the daughter cells produced from these cells retain the same feature of X chromosome inactivation, which is called X chromosome inactivation (XCI). Studying the mechanisms of XCI is important for understanding epigenetic that plays an important role in age-associated diseases. The previous studies have demonstrated that binding of RNAs and DNAs may play a role in activating gene expression. In this paper, our study aims to explore whether the mechanisms of XCI involve the RNA binding strength to X chromosome DNAs. The bioinformatics analyses based on big data were used to analyze the simulated binding strength of RNAs (RNA binding strength) to 23 chromosomes (including X chromosome and 22 human autosomes) and the characteristics of repetitive sequences in the X-inactivation centre. The results revealed that RNA binding strength of the long arm of the X chromosome that is almost entirely inactivated in XCI was significantly lower than that of all autosomes and the short arm of X chromosome, meanwhile the RNA binding strengths of inactivation regions in X chromosome were significantly lower than that of regions escaping from XCI. Different repetitive sequence clusters within the center of XCI presented a cross distribution characteristic. To further prove whether the repetitive sequences in human X chromosome involve in XCI, we cloned long interspersed element (LINE-1, L1) and short interspersed element (Alu) from human Xq13, the center of XCI, and constructed expression vectors carrying sense-antisense combination repetitive sequences (L1s or Alus). Effects of combined L1 or combined Alu sequences on expression of EGFP reporter gene were examined in stably transfected HeLa cells, which simulates the effects of repetitive sequences located on chromosomes. The results of experiments revealed transcribed L1 repetitive sequences activated EGFP reporter gene expression, so did the Alus. The experiment results suggested repetitive sequences activated genes by interaction of transcribed RNAs and DNAs. Since the binding of RNAs and DNAs can activate gene, so the low RNA binding strength of human X chromosome may be one of reasons of XCI. The cross distribution characteristics of different repetitive sequence clusters leading to a cascade of gene activation or gene inactivation may be the reason of transcriptional silencing one of the X chromosomes in female mammals.

Multi-receiver encryption enables a sender to encrypt a message and transmit the ciphertext to a set of authorized users, while no one out this group of authorized users can decrypt the message. Multi-receiver encryption is of great importance in many sectors such as broadcast communication, cloud computing, wireless communications, networking applications, e-voting, lottery, and medical applications. This paper proposes an efficient multi-receiver public key encryption scheme based on Lucas sequences. The results of the computational analysis show that, projected scheme is better against renown attacks and prevailing anonymous multireceiver algorithms

Multidrug resistance (MDR) among gram-negative bacteria is a global public health threat because it has exhausted all standard therapeutic options. This has necessitated clinicians to reconsider colistin, an “old sort” antibiotic which was deemed toxic for clinical use. Unfortunately, swift resistance towards colistin by K. pneumoniae has been reported in different parts of the world. The study aimed at investigating the genetic determinants of colistin resistance in K. pneumoniae isolated from patients admitted in a tertiary care hospital in Kampala, Uganda. 31 archived multidrug resistant K. pneumoniae isolates obtained from rectal swabs of patients admitted to the Mulago Hospital ICU were retrieved. From this collection, we selected 7 colistin resistant K. pneumoniae isolates for whole genome sequencing and molecular profile analysis. In silico sequence data analyses were performed using established web-based bioinformatics tools. The sequence analysis of the genes involved in colistin resistance revealed several non-synonymous mutations in nucleotide sequences of pmrA, pmrB, phoP, phoQ and mgrB genes. Insertion transposase genes for the insertion sequences IS1 and IS5 which are highly implicated in colistin resistance via modification or inactivation of mgrB gene were identified in all isolates and were located in the chromosomal DNA at different genomic positions. No plasmid-borne mcr genes were detected. More so, the strains harbored numerous multidrug resistant genes. This study underscores the need of genomic surveillance of K. pneumoniae strains to limit further spread of colistin resistance genes to avert the potential risk of pandrug resistance. And high throughput sequencing is imperative.

In this work, we present a new way to study the genetic code mathematical and chemical structure, based on the use of mathematical computations involving some few recently designed Fibonacci-like sequences, the “seeds” (“initial conditions”) of which are chosen according to the chemical and physical data of the three amino acids serine, arginine and leucine, playing a prominent role in a recent symmetry classification scheme of the genetic code. It appears that these mathematical sequences, of the same kind as the famous Fibonacci series, apart from their usual recurrence relations, are highly interwinned by many useful linear relationships. Using these sequences and also various sums or linear combinations of them, we derive several physical and chemical quantities of interest, as the number of total coding codons, 61, obeying various degeneracy patterns, the detailed number of hydrogen atoms, the detailed number of atoms (H/CNOS) and the integer molecular mass (or nucleon number), in the side chains of the 61 “amino acids”, also in various degeneracy patterns, in agreement with those described in the literature. Also, unexpectedly but interestingly, we find, as a by-product, an accurate description of the very chemical structure of the four ribonucleotides uridine monophosphate (UMP), cytidine monophosphate (CMP), adenosine monophosphate (AMP) and guanosine monophosphate (GMP), the building blocks of RNA whose groupings, in three units, constitutes the triplet codons. In summary, we find a full mathematical and chemical connection with the “ideal sextet’s classification scheme” mentioned above as well as with several others, notably, the Findley-Findley-McGlynn symmetrical classification. We organize the content of the text in such a way that, besides the presentation of several new research results, it has also an educational dimension. The paper could therefore be read and the computations easily worked out, also by non-experts with mathematical background.

Disorders of swallowing often lead to pneumonia when material enters the airways (aspiration). Flexible Endoscopic Evaluation of Swallowing (FEES) plays a key role in the diagnostics of aspiration but is prone to human errors. An AI-based tool could facilitate this process. Recent non-endoscopic/non-radiologic attempts to detect aspiration using machine-learning approaches have led to unsatisfying accuracy and show black box characteristics. Hence, for clinical users it is hard to trust in these model decisions. Our aim is to introduce an explainable artificial intelligence (XAI) approach to detect aspiration in FEES. Our approach is to teach the AI about the relevant anatomical structures like the vocal cords and the glottis based on 92 annotated FEES videos. Simultaneously, it is trained to detect bolus that passes the glottis and becomes aspirated. During testing, the AI successfully recognized glottis and vocal cords, but could not yet achieve satisfying aspiration detection quality. Albeit detection performance has to be optimized, our architecture results in a final model that explains its assessment by locating meaningful frames with relevant aspiration events and by highlighting the suspected bolus. In contrast to comparable AI tools, our framework is verifiable, interpretable and therefor accountable for clinical users.

Two new species of genus Piper L. from Madagascar: Piper malgassicum Papini, Palchetti, M. Gori & Rota Nodari and Piper tsarasotrae Papini, Palchetti, M. Gori & Rota Nodari, were analyzed to investigate their phylogenetic position and evolutionary history. Both plastidial and nuclear markers were used for sequencing. The plastidial markers (ndhF and trnL intron) showed a close relationship between the two species with respect to the other species of Piper. Both species appeared phylogenetically related to the African P. guineense and the Malagasian/Mascarenhas endemic P. borbonense. The nuclear marker (G3pdh) amplification produced two separate sets of sequences: “long” sequences and “short” sequences, characterized by some long deletions. Analyzing together the nuclear sequences, we observed that the “long” sequence of P. tsarasotrae had a stricter relationship to the African accessions of P. guineense, while the accession of P. malgassicum was more strictly related to P. borbonense. On the contrary both “short” sequences of P. malgassicum and P. tsaratsotrae resulted phylogenetically related to Asian accessions and more distantly related to the formerly cited species. This unexpected result was tentatively explained with a more ancient hybridization event between an ancestor of P. malgassicum and P. tsarasotrae (and possibly P. borbonense) and an Asian species of Piper. The Asian contribution would have produced the ancestors carrying the “short” sequences. A more recent hybridization event would have led to the separation of P. malgassicum from P. tsarasotrae with an African pollen-derived genome contribution from P. guineense or, more probably, an ancestor thereof, to an ancestor of P. tsarasotrae. The chromosome numbers of P. tsarasotrae (2n = about 38) and P. malgassicum (2n = about 46), were more similar to the Asian species than to the American species. Unfortunately, no chromosome number of the African species P. guineense is currently available, to compare the chromosomal numbers.

In Southeast Asia, the incidence of Leukemia, a malignant blood cancer originating from hema-topoietic progenitor cells, is on the rise, marked by a concerning 54% mortality rate. This study focuses on enhancing early-stage prediction to improve patient recovery prospects significantly. Leveraging Machine Learning and Data Science, we employ protein sequential data from frequently mutated genes such as BCL2, HSP90, PARP, and RB to predict Chronic Myeloid Leukemia (CML). Our approach relies on robust feature extraction techniques, namely Di-peptide Composition (DPC), Amino Acid Composition (AAC), and Pseudo amino acid composition (Pse-AAC), with prior attention to addressing outliers and validating feature selection through the Pearson Corre-lation Coefficient. Data augmentation ensures a well-rounded dataset for analysis. Employing a range of Machine Learning models, including Support Vector Machine (SVM), XGBoost, Random Forest (RF), K Nearest Neighbor (KNN), Decision Tree (DT), and Logistic Regression (LR), we achieve accuracy rates spanning from 66% to 94%. These classifiers undergo comprehensive as-sessment using performance metrics such as accuracy, sensitivity, specificity, F1-score, and the confusion matrix. Our proposed solution, encompassing a user-friendly web application dashboard, presents an invaluable tool for early CML diagnosis with profound implications for practitioners, offering a deploy-able asset within healthcare institutions and hospitals.

Surgical knots are sequences of half-knots (H) or half-hitches (S), defined by their number of throws, by an opposite or similar rotation compared with the previous one and for half-hitches, whether sliding (s) or blocking (b). Opposite rotation results in (more secure) symmetric (s) knots, similar rotation in asymmetric (a) knots, and changing active and passive ends has the same effect as changing rotation. Loop security is the force to keep tissue together after a first half-knot or sliding half-hitches. With polyfilament sutures H2, H3, SSs and SSsSsSs have a loop security of 10, 18, 28 and 48 Newton (N), respectively. With monofilament sutures, they are only 7, 16, 18 and 25N. Since many knots can reorganise, the definition of knot security as the force at which the knot opens or the suture breaks should be replaced by the clinically more relevant percentage of clinically dangerous and insecure knots. Secure knots with polyfilament sutures require a minimum of 4 or 5 throws, but the risk of destabilisation is high. With monofilament sutures, only 2 symmetric+4 asymmetric blocking half-hitches are secure. In conclusion, half-hitch sequences are recommended since mandatory for monofilament sutures, adding flexibility for loop security with less risk of destabilisation.

Next Generation Sequencing (NGS), now widely used in the clinical setting, offers an efficient and comprehensive molecular approach for patients with Familial hypercholesterolemia (FH). Although the dominant form of disease is mostly due to low-density lipoprotein receptor (LDLR) small-scale pathogenic variants, approximately 10% of molecularly defined FH cases are due to Copy Number Variations (CNVs). Here, we report a novel large deletion of the LDLR gene involving exons 4–18, identified by bioinformatic analysis of NGS data in an Italian family. A long PCR strategy was employed for breakpoint region analysis where an insertion of 6 nucleotides (TTCACT) was found. Two Alu sequences, identified within intron 3 and exon 18, could underlie the identified rearrangement by a Nonallelic Homologous Recombination (NAHR) mechanism. NGS proves to be a powerful tool used to precisely identify CNVs, in addition to small-scale variants in FH-related genes. At this purpose, the use and implementation of this cost-effective, efficient molecular approach meets the clinical need for personalized diagnosis in FH cases.

One of the most significant developments in the sports world over the last two decades has been the use of mathematical methods in conjunction with the massive amounts of data now available to analyze performances, identify trends and patterns, and forecast results. Football analytics has advanced significantly in recent years and continues to evolve as it becomes a more recognized and integral part of the game. Football analytics is also used to forecast game outcomes, allowing bettors to make educated guesses. This article describes mathematical concepts related to football analytics that enable a better betting strategies. We explain how the pitch is partitioned into different zones and we define possession sequences. Furthermore, we explain what an expected goals model is and which expected goals model we use in this research. Furthermore, we define two general characteristics of a player evaluation method, each corresponding to one of the equations of the Dawson model. Based on these characteristics, we describe the developments of several general approaches for evaluating players in the context of the Dawson model.

Pneumonia cases of unknown etiology in Wuhan, China, were reported to the WHO on 31st of December 2019. Later the pathogen was reported to be a novel coronavirus designated Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) that causes Coronavirus Disease 2019 (COVID-19). SARS-CoV-2 is a novel pathogenic beta coronavirus that infects humans causing severe respiratory illness. However, multifarious factors can contribute to the susceptibility to COVID-19 related morbidity and mortality such as age, gender and underlying comorbidities. Importantly, SARS-CoV and SARS-CoV-2 entry into the host cells is mediated via ACE2 receptor. However, ACE2 receptor binding affinity to SARS-CoV-2 is 4 folds higher than that to SARS-CoV. Identification of different aspects such as binding affinity, differential antigenic profiles of spike glycoproteins, and ACE2 polymorphisms might influence the investigation of potential therapeutic strategies targeting SARS-CoV-2/ACE2 binding interface. Here we aim to elaborate on SARS-CoV-2 S1/ACE2 ligand that facilitates viral internalization as well as to highlight the differences between SARS-CoVs binding affinity to ACE2. We also discuss the possible immunogenic sequences of spike glycoprotein and the effect of ACE2 polymorphism on viral binding/infectivity and host susceptibility to disease. Furthermore, targeting of ACE2 will be discussed to understand its role in therapeutics.

For any odd positive integer $x$, define $(x_n)_{n\geqslant 0} $ and $(a_n )_{n\geqslant 1} $ by setting $x_{0}=x, \,\, x_n =\cfrac{3x_{n-1} +1}{2^{a_n }}$ such that all $x_n $ are odd. The 3x+1 problem asserts that there is an $x_n =1$ for all $x$. Usually, $(x_n )_{n\geqslant 0} $ is called the trajectory of $x$. In this paper, we concentrate on $(a_n )_{n\geqslant 1} $ and call it the E-sequence of $x$. The idea is that, we consider any infinite sequence $(a_n )_{n\geqslant 1} $ of positive integers and call it an E-sequence. We then define $(a_n )_{n\geqslant 1} $ to be $\Omega-$convergent to $x$ if it is the E-sequence of $x$ and to be $\Omega-$divergent if it is not the E-sequence of any odd positive integer. We prove a remarkable fact that the $\Omega-$divergence of all non-periodic E-sequences implies the periodicity of $(x_n )_{n\geqslant 0} $ for all $x_0$. The principal results of this paper are to prove the $\Omega-$divergence of several classes of non-periodic E-sequences. Especially, we prove that all non-periodic E-sequences $(a_n )_{n\geqslant 1} $ with $\mathop {\overline {\lim } }\limits_{n\to \infty } \cfrac{b_n }{n}>\log _23$ are $\Omega-$divergent by using the Wendel's inequality and the Matthews and Watts's formula $x_n =\cfrac{3^n x_0 }{2^{b_n }}\prod\limits_{k=0}^{n-1} {(1+\cfrac{1}{3x_k })} $, where $b_n =\sum\limits_{k=1}^n {a_k } $. These results present a possible way to prove the periodicity of trajectories of all positive integers in the 3x + 1 problem and we call it the E-sequence approach.

The philosophy of testing statistical hypothesis is a natural consequence and functional extension of mathematical analysis of Probability. Along with the concept of recurrence when applied to random sequences and functions, it leads to the analysis of a priori and posterior which implies testing statistical hypothesis. Testing statistical hypothesis also involves algebraic, functional and dimensional considerations, which are found in the works of Laplace. Aspects of mathematical analysis such as universality of solutions, Laws of Large Numbers, Entropy, Information, and various functional dependencies are the main factors explained in the five properties that lead to implication of testing statistical hypothesis. Various interesting examples with modern scientific significance from genetics, astrophysics, and other areas give methodological access to answers of different problems and phenomena which are involved in the logic of testing statistical hypothesis.

Financial time series and other human-driven, non-natural processes are known to exhibit fat-tailed outcome distributions. That is, such processes demonstrate a greater tendency for extreme outcomes than the normal distribution or other natural distributional processes would predict. We examine the mathematical expectation, or simply "expectation," traditionally the probability-weighted outcome, regarded since the seventeenth century as the mathematical definition of "expectation." However, when considering the "expectation" of an individual confronted with a finite sequence of outcomes, particularly existential outcomes (e.g., a trader with a limited time to perform or lose his position in a trading operation), we find this individual "expects" the median terminal outcome over those finite trials, with the classical seventeenth-century definition being the asymptotic limit as trials increase to this. Since such finite-sequence "expectations" often differ in values from the classic one, so do the optimal allocations (e.g., growth-optimal). We examine these for fat-tailed distributions. The focus is on implementation, and the techniques described can be applied to all distributional forms.

The article presents the author's results of studying hidden rules of structural organizations of long DNA sequences in eukaryotic and prokaryotic genomes. The results concern some rules of percentages (or probabilities) of n-plets in genomes. To reveal such rules, the author considers genomic DNA nucleotide sequences as multilayers sequences of n-plets and studies the percentage contents of n-plets in different layers. Unexpected rules of invariance of total sums of percentages in certain tetra-groupings of n-plets in different layers of genomic DNA sequences are revealed. These discovered rules are candidates for the role of universal genomic rules. A tensor family of matrix representations of interrelated DNA-alphabets of 4 nucleotides, 16 doublets, 64 triplets, and 256 tetraplets is used in the study. This matrix approach allows revealing algebraic properties of the mentioned genetic rules of probabilities, which are useful for developing algebraic and quantum biology. Some analogies of the discovered genetic phenomena with phenomena of Gestalt psychology are noted and discussed. The author connects the received results about the genomic percentages rules with a supposition of P. Jordan, who is one of the creators of quantum mechanics and quantum biology, that life's missing laws are the rules of chance and probability of the quantum world.

Ostreococcus tauri is a simple unicellular green alga representing an ecologically important group of phytoplankton in oceans worldwide. Modern molecular techniques must be developed in order to understand the mechanisms that permit adaptation of microalgae to their environment. We present for the first time in O. tauri a detailed characterization of individual genomic integration events of foreign DNA of plasmid origin after PEG-mediated transformation. Vector integration appears to be random, occurring mainly at a single locus, and thus confirming the utility of this technique for insertional mutagenesis. While the mechanism of double-stranded DNA repair in the O. tauri model remains to be elucidated, we clearly demonstrate by genome resequencing that the integration of the vector leads to frequent structural variations (deletions/insertions and duplications) and some chromosomal rearrangements in the genome at the insertion loci, and often within the vector sequence itself. From these observations, we speculate that a non-homologous end joining-like mechanism is required during random insertion events, as described in plants and other freshwater algal models. PEG-mediated transformation is therefore a promising molecular biology tool, not only for functional genomic studies, but also for biotechnological research in ecologically important marine algae.

The article is devoted to biological models using recurrence sequences, which are connected with the harmonic progression 1, 1/2, …, 1/n, and some cooperative properties of genomes. The harmonic progression is itself one of the recurrence sequences based on the harmonic mean. This progression appears in the hyperbolic rules of oligomer cooperative organization in eukaryotic and prokaryotic genomes. This allows thinking that the harmonic progression is also related to inherited physiological systems, which must be structurally consistent with the genetic coding system for their transmission to descendants and survival in evolution. The harmonic progression is one of historically known mathematical series, whose features were studied by Pythagoras, Leibniz, Newton, Euler, Fourier, Dirichlet, Riemann. It is widely used in many known algorithms and is closely related to some other important mathematical objects, for example, the function of the natural logarithm and harmonic numbers. Accordingly, the article describes the possibilities of using these interrelated mathematical objects to model biological structures, including logarithmic spirals and some other. Modeling inherited spiral configurations seems to be a particularly urgent task, since they are extremely common at all levels of organization of living bodies and, according to Goethe, are lines of life. The principle of a recurrence similarity, that is a special similarity of parts and transformations presented in recurrence sequences of numbers and matrix operators (the scale similarity and scale transformations are only particular cases of such similarity), is considered as one of the key principles of structural organization of living bodies.

The loop and knot securities of 2 braided and 2 monofilament sutures of four diameters (3.0, 2.0, 0, 1) were evaluated with a tensiometer for four 4-throw knots known to be secure with a 2.0 polyfilament suture. Loop security of monocryl 1 is low, being 14.7 + 3.0 Newton (N) for a 3-throw half-knot (H3) and 15.4 + 2.4 N and 28.3+ 10 N for 2 (SSs) and 4-throw (SSsSsSs) symmetrical sliding half-hitches. This is lower than 18, 24, and 46 N for similar knots with vicryl. Polyfilament sutures have excellent knot security for all diameters. Occasionally, some slide to open with slightly lower knot security, especially for larger diameters, although not clinically problematic. Knot se-curity of monofilament sutures was unpredictable for all 4 knots, especially for larger diameters, resulting in many clinically insecure knots. A secure monofilament knot requires a 6-throw knot with 2 symmetrical sliding half-hitches or 2 symmetrical half-knots secured with 4 asymmetric blocking half-hitches. In conclusion, with polyfilament sutures, 4 or 5-throw half-knot or half-hitch sequences result in secure knots. For monofilament sutures, loop and knot security is much less, half-knot combinations should be avoided, and secure knots require 6-throw knots with 4 asym-metric half hitches. This conclusion is valid for gynecologic laparoscopic and open surgery, and for surgery in general.

Viruses frequently contain overlapping genes, which encode functionally unrelated proteins from the same DNA or RNA region but in different reading frames. Yet overlapping genes are often overlooked during genome annotation, in particular in DNA viruses. Here we looked for the presence of overlapping genes likely to encode a functional protein in human parvovirus B19 (genus erythroparvovirus), using an experimentally validated software, Synplot2. Synplot2 detected an open reading frame, X, conserved in all erythroparvoviruses, which overlaps the VP1 capsid gene, and is under highly significant selection pressure. In a related virus, human parvovirus (genus tetraparvovirus), Synplot2 also detected an open reading frame under highly significant selection pressure, ARF1, which overlaps the VP1 gene. X and ARF1 have exactly the same location (both overlap the region of VP1 encoding the phospholipase A2 domain), and encode proteins with similar predicted properties, such as a transmembrane region, strongly suggesting that they are homologous. These findings provide compelling evidence that the X protein must be expressed and functional. It is probably translated either from a polycistronic mRNA by a non-canonical mechanism, or from an unmapped monocistronic mRNA. Finally, we also discovered proteins predicted to be expressed from a frame overlapping VP1 in other species related to parvovirus B19: porcine parvovirus 2 (Z protein) and bovine parvovirus 3 (X-like protein).

The aim of this paper is to introduce a new methodology for the fault diagnosis of induction machines working in transient regime, when time-frequency analysis tools are used. The proposed method relies on the use of the optimized Slepian window for performing the short time Fourier transform (STFT) of the stator current signal. It is shown that for a given sequence length of finite duration the Slepian window has the maximum concentration of energy, greater than can be reached with a gated Gaussian window, which is usually used as analysis window. In this paper the use and optimization of the Slepian window for fault diagnosis of induction machines is theoretically introduced and experimentally validated through the test of a 3.15 MW induction motor with broken bars during the start-up transient. The theoretical analysis and the experimental results show that the use of the Slepian window can highlight the fault components in the current's spectrogram with a significant reduction of the required computational resources.

This paper presents a novel approach to demonstrate the Collatz Conjecture, an unsolved problem in mathematics stating that all positive integers will eventually converge to 1 when subjected to a recursive sequence of operations. We introduce Algebraic Inverse Trees (AITs), innovative data structures that characterize relationships within the Collatz sequence by tracing inverse transformations back to the origin. Leveraging properties unveiled through this inverted representation, including absence of non-trivial cycles and guaranteed convergence of paths, we construct a topological framework to formally prove that all trajectories invariably lead to 1 under the Collatz iterative process. This modern technique provides not only the machinery to manifest a rigorous proof, but also grants intuitive insights into the deep complexity of this notorious yet elementary conjecture.

Loess provides a valuable terrestrial record of past environmental conditions, including the dynamics of air mass circulation responsible for dust transport. Here we explore variations in the luminescence characteristics of sedimentary quartz and feldspar – dominant minerals in loess-palaeosol sequences (LPS) - as possible tools for identifying changes in source. We investigate luminescence sensitivity, a rapidly measurable index which is the product of interplay between source lithology and the history of the mineral in question. Variations in sensitivity down profile may therefore reflect, among other factors such as pedogenesis and reworking, changes in sediment provenance. We undertake an empirical investigation of the luminescence sensitivity of quartz and feldspar from different grain-size fractions from the climatically sensitive Schwalbenberg LPS in the German Rhine valley, comparing samples from a 30 m core spanning the last full glacial cycle with samples of oxygen isotope stage (OIS) 3-2 age exposed within a c. 6 m profile downslope. The temporal overlap enables comparison of luminescence characteristics with respect to possible provenance change during that timeframe. We find an overall inverse relationship between quartz and feldspar sensitivity, as well as variability in sensitivity between different quartz grain sizes. There is some indication that feldspar sensitivity increases during periods of soil formation down the core. In particular, measurements of IR50 sensitivity on unprocessed sediments show correlation with down-profile trends in more established indicators of provenance. This suggests it may be used to provide a reliable, rapid scan of sensitivity changes, and may suggest source variability over millennial timescales.

The author's method of oligomer sums for analysis of oligomer compositions of eukaryotic and prokaryotic genomes is described. The use of this method revealed the existence of general rules for cooperative oligomeric organization of a wide list of genomes. These rules are called hyperbolic because they are associated with hyperbolic sequences including the harmonic progression 1, 1/2, 1/3, .., 1/n. These rules are demonstrated by examples of quantitative analysis of many genomes from the human genome to the genomes of archaea and bacteria. The hyperbolic (harmonic) rules, speaking about the existence of algebraic invariants in full genomic sequences, are considered as candidates for the role of universal rules for the cooperative organization of genomes. The described phenomenological results were obtained as consequences of the previously published author's quantum-information model of long DNA sequences. The oligomer sums method was also applied to the analysis of long genes and viruses including the COVID-19 virus; this revealed, in characteristics of many of them, the phenomenon of such rhythmically repeating deviations from model hyperbolic sequences, which are associated with DNA triplets. In addition, an application of the oligomer sums method are shown to the analysis of the following long sequences: 1) amino acid sequences in long proteins like the protein Titin; 2) phonetic sequences of long Russan literary texts (for checking of thoughts of many authors that phonetic organization of human languages is deeply connected with the genetic language). The topics of the algebraic harmony in living bodies and of the quantum-information approach in biology are discussed.

In 1937,German mathematician L.Collatz proposed a conjeture:for any positive integer,if it is even,divide it by 2;if it is odd,multiply it by 3 and 1 to get an even number;keep going on and on, the final result is 1.This paper gives a sequence classification of odd numbers,and gives 1-1 mapping of odd numbers and odd number sequences,and uses these results to give a proof of this conjecture.