Pseudo-random sequences of non-maximum length on shift registers with reducible and primitive polynomials

Inhomogeneous pseudo-random sequences of non-maximal length formed by shift registers with linear feedbacks based on a characteristic polynomial of degree n of the form ϕ(x)=ϕ1(x)ϕ2(x), where ϕ1(x) = x m1 ⊕ 1, and ϕ2(x) of degree m 2 is primitive (m 1 = 2 k , k is a positive integer, n = m 1 + m 2) are considered. Three schemes that are equivalent in terms of periodic sequence structures were considered. Of the greatest interest are the shift registers connected in an arbitrary way using a modulo-two adder, the feedbacks in which correspond to the multipliers ϕ1(x) and ϕ2(x) the polynomials ϕ(x). In this case, there is a complex process of forming output sequences, which involves both direct and inverse M-sequences. The statement about the singularity of the generated sequences at m 1 = 4 is proved, which is confirmed by their decimation with an index equal to the period of the primitive polynomial.

On the generative capacity of matrix insertion-deletion systems of small sum-norm

A matrix insertion-deletion system (or matrix ins-del system) is described by a set of insertion-deletion rules presented in matrix form, which demands all rules of a matrix to be applied in the given order. These systems were introduced to model very simplistic fragments of sequential programs based on insertion and deletion as elementary operations as can be found in biocomputing. We are investigating such systems with limited resources as formalized in descriptional complexity. A traditional descriptional complexity measure of such a matrix ins-del system is its size $$s=(k;n,i',i'';m,j',j'')$$ s = ( k ; n , i ′ , i ′ ′ ; m , j ′ , j ′ ′ ) , where the parameters from left to right represent the maximal matrix length, maximal insertion string length, maximal length of left contexts in insertion rules, maximal length of right contexts in insertion rules; the last three are deletion counterparts of the previous three parameters. We call the sum $$n+i'+i''+m+j'+j''$$ n + i ′ + i ′ ′ + m + j ′ + j ′ ′ the sum-norm of s. We show that matrix ins-del systems of sum-norm 4 and sizes (3; 1, 0, 0; 1, 2, 0), (3; 1, 0, 0; 1, 0, 2), (2; 1, 2, 0; 1, 0, 0), (2; 1, 0, 2; 1, 0, 0), and (2; 1, 1, 1; 1, 0, 0) describe the recursively enumerable languages. Moreover, matrix ins-del systems of sizes (3; 1, 1, 0; 1, 0, 0), (3; 1, 0, 1; 1, 0, 0), (2; 2, 1, 0; 1, 0, 0) and (2; 2, 0, 1; 1, 0, 0) can describe at least the regular closure of the linear languages. In fact, we show that if a matrix ins-del system of size s can describe the class of linear languages $$\mathrm {LIN}$$ LIN , then without any additional resources, matrix ins-del systems of size s also describe the regular closure of $$\mathrm {LIN}$$ LIN . Finally, we prove that matrix ins-del systems of sizes (2; 1, 1, 0; 1, 1, 0) and (2; 1, 0, 1; 1, 0, 1) can describe at least the regular languages.

A study on the association between accessory maxillary ostium and maxillary sinus mucosal thickening using cone beam computed tomography

Background Accessory maxillary ostium (AMO) has a major role to play in the aetiology of maxillary sinusitis. Mucosal thickening is one of the key radiographic features of chronic maxillary sinusitis. The aim of this study was to identify the location of the AMOs and investigate the association between Mucosal Thickening [MT] and AMO using Cone Beam Computed Tomography [CBCT]. Methods CBCT scans of 400 maxillary sinuses from the records of 200 patients who seeked various dental treatments at the Thumbay Dental Hospital, Gulf Medical University, Ajman, United Arab Emirates were evaluated. The incidence, anatomical position and maximal length of accessory maxillary ostia (AMO) in the maxillary antrum were reviewed using CBCT by two examiners. The association between MTs and AMOs were also analysed. Results Among the 200 CBCT scans, 131 belonged to male patients and 69 scans belonged to female subjects within the age group of 18–65 years (mean age 41.32 years). AMOs were found in 142 maxillary antra (35.5 %). The inter-observer reliability for using CBCT to detect AMO was (k = 0.83). There was no significant difference in the frequency of AMOs when the age (P = 0.19) and gender (P = 0.54) distribution were considered. Sinuses with AMOs, showed significantly greater frequency of MTs (p = 0.001). AMOs with maximal length of less than 1mm were most commonly observed (51.40 %). AMOs with larger greater maximal length were associated with higher degrees of MT. The location of the AMOs, were not affected by the degree of MT. Conclusions The study demonstrates a clear association between degree of MT and occurrence of AMO in the maxillary sinus. However, the location of the AMO is independent of the degree of the MT. There is a greater probability of finding an AMO in the maxillary sinus if the MT in the sinus is more than 3 mm.

Decidability Questions for Insertion Systems and Related Models

Insertion systems or insertion grammars are a generative formalism in which words can only be generated by starting with some axioms and by iteratively inserting strings subject to certain contexts of a fixed maximal length. It is known that languages generated by such systems are always context sensitive and that the corresponding language classes are incomparable with the regular languages. On the other hand, it is possible to generate non-semilinear languages with systems having contexts of length two. Here, we study decidability questions for insertion systems. On the one hand, it can be seen that emptiness and universality are decidable. Moreover, the fixed membership problem is solvable in deterministic polynomial time. On the other hand, the usually studied decidability questions such as, for example, finiteness, inclusion, equivalence, regularity, inclusion in a regular language, and inclusion of a regular language turn out to be undecidable. Interestingly, the latter undecidability results can be carried over to other models which are basically able to handle the mechanism of inserting strings depending on contexts. In particular, new undecidability results are obtained for pure grammars, restarting automata, clearing restarting automata, and forgetting automata.

Thermodynamics of Ideal Gas at Planck Scale with Strong Quantum Gravity Measurement

More recently in J. Phys. A: Math. Theor. 53, 115303 (2020), we have introduced a set of noncommutative algebra that describes the space-time at the Planck scale. The interesting significant result we found is that the generalized uncertainty principle induced a maximal length of quantum gravity which has different physical implications to the one of generalized uncertainty principle with minimal length. The emergence of a maximal length in this theory revealed strong quantum gravitational effects at this scale and predicted the detection of gravity particles with low energies. To make evidence of these predictions, we study the dynamics of a free particle confined in an infinite square well potential in one dimension of this space. Since the effects of quantum gravity are strong in this space, we show that the energy spectrum of this system is weakly proportional to the ordinary one of quantum mechanics free of the theory of gravity. The states of this particle exhibit proprieties similar to the standard coherent states which are consequences of quantum fluctuation at this scale. Then, with the spectrum of this system at hand, we analyze the thermodynamic quantities within the canonical and micro-canonical ensembles of an ideal gas made up of N indistinguishable particles at the Planck scale. The results show a complete consistency between both statistical descriptions. Furthermore, a comparison with the results obtained in the context of minimal length scenarios and black hole theories indicates that the maximal length in this theory induces logarithmic corrections of deformed parameters which are consequences of a strong quantum gravitational effect.

Recognition of Tibetan Maximal-length Noun Phrases Based on Syntax Tree

Frequently corresponding to syntactic components, the Maximal-length Noun Phrase (MNP) possesses abundant syntactic and semantic information and acts a certain semantic role in sentences. Recognition of MNP plays an important role in Natural Language Processing and lays the foundation for analyzing and understanding sentence structure and semantics. By comparing the essence of different MNPs, this article defines the MNP in the Tibetan language from the perspective of syntax tree. A total of 6,038 sentences are extracted from the syntax tree corpus, the structure type, boundary feature, and frequency of MNPs are analyzed, and the MNPs are recognized by applying the sequence tagging model and the syntactic analysis model. The accuracy, recall, and F1 score of the recognition results of applying sequence tagging model are 87.14%, 84.72%, and 85.92%, respectively. The accuracy, recall, and F1 score of the recognition results of applying syntactic analysis model are 87.66%, 87.63%, and 87.65%, respectively.