Olga Taussky-Todd suggested the problem of determining the possible values of integer circulant determinants. To solve a special case of the problem, Laquer gave a factorization of circulant determinants. In this paper, we give a modest generalization of Laquer's theorem. Also, we give an application of the generalization to integer group determinants.

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The determiners are analyzed often in the midst of nominal group. They are sometimes believed as “devoid of sense”, nevertheless, these elements have a fundamental role in the interpretation of an utterance. They can combine with each other according to precise possibilities and so on make a real syntactic group. The determinant group can have a complex structure and it must be considered as an independent syntactic unit (syntagm). In this paper we will analyze and compare the structure of determinant group in French and Persian languages. French: Analyse comparee de la structure du groupe determinant du français et du person Les determinants (vabasteha-ye pisin-e esm) sont etudies ici plutot au sein du groupe nominal. On les croit parfois "depourvus de sens", pourtant, ils jouent un role fondamental dans l’interpretation d’un enonce. Ils peuvent se combiner entre eux selon des possibilities combinatoires precises et former ainsi un veritable groupe. Le groupe determinant peut avoir une structure assez complexe et doit être considere comme une unite syntaxique (syntagme) independante. Nous avons pour objectif, dans cet article, la definition des structures du groupe determinant du français et du persan. Ainsi, nous analyserons et comparerons les constructions possibles de ce groupe dans ces deux langues.Keywords: Determinant, Groupe Determinant, Analyse Syntagmatique, Structure Syntaxique, Etude Comparee Français-Persan

In traditional Data Envelopment Analysis (DEA) approaches, inputs and outputs are usually considered as exact and real values. The relative efficiency of the Decision Making Units (DMUs) is evaluated and it is known that the factors are inputs and/or outputs. However, there are some conditions under which the efficiency of DMUs should be calculated while the data are integer and ambiguous. Therefore, various integer DEA models have been proposed to determine the performance of DMUs when integer data and fuzzy factors are available. In addition, there are cases where the efficiency of DMUs should be determined when integer data and flexible factors are available. Therefore, some integer DEA methods have been proposed to calculate the performance of DMUs and specify the role of flexible measures when some of the data are integer and flexible factors are available. However, there are some situations where there are integer data, fuzzy integer measures and flexible factors. Therefore, this paper sheds light on the nature of the model to determine the efficiency of DMUs when there are integer inputs and/or outputs, flexible factors and fuzzy integer measures, and determines the role of factors with uncertain inputs or outputs. In fact, slacks are addressed and a slack-based efficiency measure (SBM) is defined to compute the performance of DMUs in the presence of flexible factors, integer data and fuzzy integer measures. The proposed approaches are demonstrated and illustrated using an example.

‎Sombor index of a graph $G$ is defined by $SO(G) = \sum_{uv \in E(G)} \sqrt{d^2_G(u)+d^2_G(v)}$, where $d_G(v)$ is the degree of the vertex $v$ of $G$. An $r$-degree graph is a graph whose degree sequence includes exactly $r$ distinctive numbers. In this article, we study $r$-degree connected graphs with integer Sombor index for $r \in \{5, 6, 7\}$. We show that: if $G$ is a 5-degree connected graph of order $n$ with integer Sombor index then $n \geq 50$ and the equality occurs if only if $G$ is a bipartite graph of size 420 with $SO(G) = 14830$; if $G$ is a 6-degree connected graph of order $n$ with integer Sombor index then $n \geq 75$ and the equality is established only for the bipartite graph of size $1080$; and if $G$ is a 7-degree connected graph of order $n$ with integer Sombor index then $n \geq 101$ and the equality is established only for the bipartite graph of size $1680$.

Privacy issues during data publishing is an increasing concern of involved entities. The problem is addressed in the field of statistical disclosure control with the aim of producing protected datasets that are also useful for interested end users such as government agencies and research communities. The problem of producing useful protected datasets is addressed in multiple computational privacy models such as k-anonymity in which data is clustered into groups of at least k members. Microaggregation is a mechanism to realize k-anonymity. The objective is to assign records of a dataset to clusters and replace the original values with their associated cluster centers which are the average of assigned values to minimize information loss in terms of the sum of within group squared errors (SSE). While the problem is shown to be NP-hard in general, there is an optimal polynomial-time algorithm for univariate datasets. This paper shows that the assignment of the univariate microaggregation algorithm cannot produce optimal partitions for integer observations where the computed centroids have to be integer values. In other words, the integrality constraint on published quantities has to be addressed within the algorithm steps and the optimal partition cannot be attained using only the results of the general solution. Then, an effective method that considers the constraint is proposed and analyzed which can handle very large numerical volumes. Experimental evaluations confirm that the developed algorithm not only produces more useful datasets but also is more efficient in comparison with the general optimal univariate algorithm.