In this paper, we extend the BT inverse to the set of third-order tensors, and we call it the T-BT inverse. We give characterizations and properties of the inverse by applying tensor decomposition. ...Based on the inverse, we introduce a new binary relation: T-BT order. Furthermore, by applying the T-BT order, we introduce a generalized core partial order (called T-GC partial order).
In the last years, several studies have investigated the role of technology in teaching and learning mathematics. However, the specific role of computer algebra systems (CAS) in early algebra in ...contrast to graphic calculators (GC) is still unclear. The CAYEN project is researching this field by comparing 13-year-old pupils—one GC class and two CAS classes have been observed while acquiring elementary algebraic competences with nearly the same teaching sequence. The field of algebraic competences is split into syntactic abilities and symbol sense. The results of this explorative case study show that the development of symbol sense is influenced by the adoption of CAS in the learning process. Especially when transitioning from arithmetic to algebra, the pupils’ views of algebra as well as their conceptions of algebraic objects seem to be affected by the availability of CAS.
Computational techniques for analyzing biological images offer a great potential to enhance our knowledge of the biological processes underlying disorders of the nervous system. Friedreich’s Ataxia ...(FRDA) is a rare progressive neurodegenerative inherited disorder caused by the low expression of frataxin, which is a small mitochondrial protein. In FRDA cells, the lack of frataxin promotes primarily mitochondrial dysfunction, an alteration of calcium (Ca
2+
) homeostasis and the destabilization of the actin cytoskeleton in the neurites and growth cones of sensory neurons. In this paper, a computational multilinear algebra approach was used to analyze the dynamics of the growth cone and its function in control and FRDA neurons. Computational approach, which includes principal component analysis and a multilinear algebra method, is used to quantify the dynamics of the growth cone (GC) morphology of sensory neurons from the dorsal root ganglia (DRG) of the YG8sR humanized murine model for FRDA. It was confirmed that the dynamics and patterns of turning were aberrant in the FRDA growth cones. In addition, our data suggest that other cellular processes dependent on functional GCs such as axonal regeneration might also be affected. Semiautomated computational approaches are presented to quantify differences in GC behaviors in neurodegenerative disease. In summary, the deficiency of frataxin has an adverse effect on the formation and, most importantly, the growth cones’ function in adult DRG neurons. As a result, frataxin deficient DRG neurons might lose the intrinsic capability to grow and regenerate axons properly due to the dysfunctional GCs they build.
Recent developments in the area of process and product integration have enabled the systematic identification of suitable candidate molecules to meet certain process performance. In this approach, ...the property targets for the input molecules corresponding to the optimum process performance have been identified in the first step and the molecules that have the target properties have been designed in the next step. The focus of this work is to develop a combined property clustering and GC
+ algorithm to identify molecules that meet the property targets identified during the process design stage. In our earlier works, a methodology was introduced for identifying molecules with a given set of properties by combining property clustering and group contribution methods. Yet, there are situations when the property contributions of some of the molecular groups of interest are not available in literature. To address this limitation, an algorithm has been developed to include the property contributions predicted by combined group contribution and connectivity indices methods into the cluster space. For the design of simple monofunctional molecules, a modified visual approach has been used, while for the design of more complicated structures an algebraic method has been developed. The applicability of the algebraic method has been increased by including the property contributions from second and third order groups.
This letter exploits geometric (Clifford) algebra (GA) theory to devise and introduce a new adaptive filtering strategy. From a least-squares cost function, the gradient is calculated following ...results from geometric calculus (GC), the extension of GA to handle differential and integral calculus. The novel GA least-mean-squares (GA-LMS) adaptive filter, which inherits properties from standard adaptive filters and from GA, is developed to recursively estimate a rotor (multivector), a hypercomplex quantity able to describe rotations in any dimension. The adaptive filter (AF) performance is assessed via a 3-D point-clouds registration problem, which contains a rotation estimation step. Calculating the AF computational complexity suggests that it can contribute to reduce the cost of a full-blown 3-D registration algorithm, especially when the number of points to be processed grows. Moreover, the employed GA/GC framework allows for easily applying the resulting filter to estimating rotors in higher dimensions.
We establish an isomorphism between the Grothendieck–Teichmüller Lie algebra
grt
1
in depth two modulo higher depth and the cohomology of the two-loop part of the graph complex of internally ...connected graphs
ICG
(
1
)
. In particular, we recover all linear relations satisfied by the brackets of the conjectural generators
σ
2
k
+
1
modulo depth three by considering relations among two-loop graphs. The Grothendieck–Teichmüller Lie algebra is related to the zeroth cohomology of Kontsevich’s graph complex
GC
2
via Willwacher’s isomorphism. We define a descending filtration on
H
0
(
GC
2
)
and show that the degree two components of the corresponding associated graded vector spaces are isomorphic under Willwacher’s map.
Coding theory has several applications in genetics and bioengineering. This paper constructs codes over an alphabet {A,C,G,T} relevant to the design of synthetic DNA strands used in DNA microarrays, ...as DNA tags in chemical libraries and in DNA computing. The codes are designed to avoid unwanted hybridizations and to ensure uniform melting temperatures. Specifically, the codes considered here satisfy a Hamming distance constraint and a GC-content constraint. In comparison with previous work, longer codes are constructed, the examination of cyclic and extended cyclic codes is more comprehensive, attention is paid to the mapping from field or ring elements to {A,C,G,T}, cosets of codes are used and a nonlinear shortening operation is performed. Many new best codes are constructed, and are reproducible from the information presented here.
In the present paper, we introduce the notion of a $ \mathcal{C}^{\star} $-algebra valued bipolar metric space and prove coupled fixed point theorems. Some of the well-known outcomes in the ...literature are generalized and expanded by the results shown. An example and application to support our result is presented.
Let R be a Noetherian ring and let C be a semidualizing R-module. In this paper, we impose various conditions on C to be dualizing. For example, as a generalization of Xu
21
, Theorem 3.2, we show ...that C is dualizing if and only if for an R-module M, the necessary and sufficient condition for M to be C-injective is that π
i
( ,M) = 0 for all ∈Spec (R) and all i≠ht ( ), where π
i
is the invariant dual to the Bass numbers defined by Enochs and Xu
8
.
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