Quantum multipartite entangled states play significant roles in quantum information processing. By using difference schemes and orthogonal partitions, we construct a series of infinite classes of ...irredundant mixed orthogonal arrays (IrMOAs) and thus provide positive answers to two open problems. The first is the extension of the method for constructing homogeneous systems from orthogonal arrays to heterogeneous multipartite systems with different individual levels. The second is the existence of
k
-uniform states in heterogeneous quantum systems. We present explicit constructions of two- and three-uniform states for arbitrary heterogeneous multipartite systems with coprime individual levels and characterize the entangled states in heterogeneous systems consisting of subsystems with nonprime power dimensions as well. Moreover, we obtain infinite classes of
k
-uniform states for heterogeneous multipartite systems for any
k
≥
2
. The non-existence of a class of IrMOAs is also proved.
Several bibliographic databases offer a free tool that enables one to determine the collaboration distance or co-authorship distance between researchers. This paper addresses a real-life application ...of the collaboration distance. It concerns somewhat unusual clustering; namely clustering in which the average distances in each cluster need to be maximised. We briefly consider a pair of clusterings in which two cluster partitions are uniform and orthogonal in the sense that in each partition all clusters are of the same size and that no pair of elements belongs to the same cluster in both partitions. We consider different objective functions when calculating the score of the pair of orthogonal partitions. In this paper the Wiener index (a graph invariant, known in chemical graph theory) is used. The main application of our work is an algorithm for scheduling a series of parallel talks at a major conference.
Finney has used orthogonal partitions in the context of the search for higher order (coarser) partitions of given Latin squares. Hedayat and Seiden use the term F-square to denote higher order ...partitions that are orthogonal to both rows and columns. This note is a short expository treatment of orthogonal partitions in general and is based on the identification of a partition with a vector subspace of Euclidean N-space RN. This identification is not new as it is part of the usual vector space approach to analysis of variance. This approach puts the concept of orthogonal partitions in a simple light unencumbered by the language of design of experiments. Another advantage is that certain published bounds on the maximum number of orthogonal partitions of specified type are immediate from the dimensionality restriction imposed by RN. In addition, some counting problems are identified which are of possible interest to researchers in design of experiments and combinatorics.
A general method of analysis is given for a design with a set of treatments added to a p-way orthogonal classification. If there is any grouping within the treatments, the treatment sum of squares ...may be partitioned; the partitioning is assisted by expressing this sum of squares as a quadratic form in the estimated treatment parameters. When the treatments fall into two groups, possibly with unequal replication, the complete partition of the treatment sum of squares is derived. Designs on three-way classifications are considered in some detail. The more useful ones are mostly on single Latin squares, and new ways of adding various numbers of treatments to 5 × 5 and 6 × 6 Latin squares are described, examples of possible designs being given. Lack of balance of the treatments with respect to one of the orthogonal classifications may be compensated for in another classification so that some designs are better balanced with three classifications than with one or two.
Zeolite AST framework is one of the most versatile cage frameworks due to its functional and structural simplicity, consisting of two four- and six-membered rings with tunable pore sizes. The AST ...framework provides an excellent model for the systematic study of the possible substitution of atoms at the centre of the primary building unit of the framework. It has recently been demonstrated to be one of the most attractive structures for deriving metal organic frameworks and zeolite-like metal organic frameworks, with adjustable large and extra-large cavities. This specific property has given tremendous thrust to the synthesis, characterization, and modification of new novel frameworks with an AST structure that are proving to be very effective materials in the fields of optics, heterogeneous catalysis, and hydrogen storage. In our study, we employ relativistic topological models that facilitate computations of exact mathematical expressions for the degree and distance based structural descriptors by using an orthogonal approach to partition the bonds of the rectangular AST framework. Such expressions would find novel applications in QSAR and QSPR models for large AST zeolites, especially in the derivations of thermodynamic, spectroscopic and topological properties of these materials.
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•Elucidated the significance of the zeolite AST structure for its applicability in molecule sieves on the basis of size, shape and polarity.•Employed an elegant graph theoretical cut method to derive exact analytical expressions for the various topological descriptors for zeolite AST materials.•Overcoming the complexity of any preprocessing steps on molecular materials, the outlined technique yields the predictive power for structure–property relations.•The structural description of zeolite AST materials that contain tunnels and cages is explored.•Topological studies considered for zeolite AST can give new impetus for the potential synthesis of new isomorphic zeotypes of industrial, medical and environmental importance.
•We re-establish a theory for the perfect reconstruction of bandlimited graph signals in the single-node aggregation sampling scheme.•An orthogonal partition selection strategy (OPSS) is proposed for ...the multi-node aggregation sampling scheme, and the corresponding reconstruction formulas are provided.•In a noisy regime, an improved OPSS (IOPSS) is proposed to guarantee a stable reconstruction by presetting a threshold.
We study the sampling of graph signals with successive local aggregations, which computes measurements as the result of iterated applications of the aggregation operator observed at one or few nodes. It has been proved in the literature that using the first few observations at a single node, a perfect reconstruction can be achieved for bandlimited graph signals. Yet, it requires some restrictions to make the sampling matrix have a Vandermonde structure. Unfortunately, Vandermonde matrices are notoriously ill-conditioned resulting in severe instability even for moderately-sized graphs. Moreover, such a sampling strategy is not applicable for the sampling scheme at multiple nodes. In view of these, we re-establish a theory for the perfect reconstruction in this paper, starting with the single-node sampling scheme. To handle large-sized graphs well, we extend it to the multi-node sampling scheme. To do this, an orthogonal partition selection strategy (OPSS) is proposed to find a node set I with a minimum size such that the observations at I guarantee a perfect reconstruction for bandlimited graph signals, and two reconstruction methods are provided. When considering a noisy regime, a performance analysis of OPSS is provided, and thus yields an improved OPSS (IOPSS) which guarantees a stable reconstruction by presetting a threshold σts. Finally, several experiments are conducted to evaluate the performance of these techniques.
In this paper, by using the Hamming distance, we establish a relation between quantum error-correcting codes ((N,K,d+1))s and orthogonal arrays with orthogonal partitions. Therefore, this is a ...generalization of the relation between quantum error-correcting codes ((N,1,d+1))s and irredundant orthogonal arrays. This relation is used for the construction of pure quantum error-correcting codes. As applications of this method, numerous infinite families of optimal quantum codes can be constructed explicitly such as ((3,s,2))s for all si≥3, ((4,s2,2))s for all si≥5, ((5,s,3))s for all si≥4, ((6,s2,3))s for all si≥5, ((7,s3,3))s for all si≥7, ((8,s2,4))s for all si≥9, ((9,s3,4))s for all si≥11, ((9,s,5))s for all si≥9, ((10,s2,5))s for all si≥11, ((11,s,6))s for all si≥11, and ((12,s2,6))s for all si≥13, where s=s1⋯sn and s1,…,sn are all prime powers. The advantages of our approach over existing methods lie in the facts that these results are not just existence results, but constructive results, the codes constructed are pure, and each basis state of these codes has far less terms. Moreover, the above method developed can be extended to construction of quantum error-correcting codes over mixed alphabets.
By using difference schemes, orthogonal partitions and a replacement method, some new methods to construct pure quantum error-correcting codes are provided from orthogonal arrays. As an application ...of these methods, we construct several infinite series of quantum error-correcting codes including some optimal ones. Compared with the existing binary quantum codes, more new codes can be constructed, which have a lower number of terms (i.e., the number of computational basis states) for each of their basis states.
This paper presents a method based on orthogonal arrays of constructing pure quaternary quantum error-correcting codes. As an application of the method, some infinite classes of quantum ...error-correcting codes with distances 2, 3, and 4 can be obtained. Moreover, the infinite class of quantum codes with distance 2 is optimal. The advantage of our method also lies in the fact that the quantum codes we obtain have less items for a basis quantum state than existing ones.
Quantum information and quantum computing are very rapidly developing research directions in recent years. This reprint introduces some of the latest developments in quantum theory, including basic ...physical theories such as quantum entanglement, and quantum information processing; and applications such as quantum secret key allocation, and several types of quantum algorithms. This reprint is suitable for some researchers or students to read.
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