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For functions f1, . . ., fn on a set D, we characterize their linear independence with an invertible matrix from their values at n distinct points in D. With the matrix, the pointwise convergence of a sequence {gk} of functions in the span{f1, · · ·, fn} is shown to be equivalent to those of the sequences of the coordinates of gks in the span. When fis are bounded, a pointwise convergent sequence {gk} must uniformly converge to a function in the span. It turns out that the limit of a convergent sequence {gk} inherits the continuity, differentiability, and integrability of fis. Furthermore the (pointwise or uniform) convergence of a sequence of solutions of an n-th order constant coefficients linear differential equation is completely determined by that of the sequence of relevant initial conditions. [ABSTRACT FROM AUTHOR]

The Thue–Morse sequence $\{t(n)\}_{n\geqslant 0}$ is the indicator function of the parity of the number of ones in the binary expansion of nonnegative integers n , where $t(n)=1$ (resp. $=0$) if the binary expansion of n has an odd (resp. even) number of ones. In this paper, we generalize a recent result of E. Miyanohara by showing that, for a fixed Pisot or Salem number $\beta>\sqrt {\varphi }=1.272019\ldots $ , the set of the numbers $$\begin{align*}1,\quad \sum_{n\geqslant1}\frac{t(n)}{\beta^{n}},\quad \sum_{n\geqslant1}\frac{t(n^2)}{\beta^{n}},\quad \dots, \quad \sum_{n\geqslant1}\frac{t(n^k)}{\beta^{n}},\quad \dots \end{align*}$$ is linearly independent over the field $\mathbb {Q}(\beta)$ , where $\varphi :=(1+\sqrt {5})/2$ is the golden ratio. Our result yields that for any integer $k\geqslant 1$ and for any $a_1,a_2,\ldots ,a_k\in \mathbb {Q}(\beta)$ , not all zero, the sequence { $a_1t(n)+a_2t(n^2)+\cdots +a_kt(n^k)\}_{n\geqslant 1}$ cannot be eventually periodic. [ABSTRACT FROM AUTHOR]

Elsner, Luca and Tachiya proved in [4] that the values of the Jacobi-theta constants θ 3 (m τ) and θ 3 (n τ) are algebraically independent over Q for distinct integers m , n under some conditions on τ . On the other hand, in [3] Elsner and Tachiya also proved that three values θ 3 (m τ) , θ 3 (n τ) and θ 3 (ℓ τ) are algebraically dependent over Q . In this article we prove the non-vanishing of linear forms in θ 3 (m τ) , θ 3 (n τ) and θ 3 (ℓ τ) under various conditions on m , n , ℓ , and τ . Among other things we prove that for odd and distinct positive integers m , n > 3 the three numbers θ 3 (τ) , θ 3 (m τ) and θ 3 (n τ) are linearly independent over Q ¯ when τ is an algebraic number of some degree greater or equal to 3. In some sense this fills the gap between the above-mentioned former results on theta constants. A theorem on the linear independence over C (τ) of the functions θ 3 (a 1 τ) , ⋯ , θ 3 (a m τ) for distinct positive rational numbers a 1 , ⋯ , a m is also established. [ABSTRACT FROM AUTHOR]

We consider the l-th order linear fractional differential equations with constant coefficients. Here l ∈ ℕ is the ceiling for the highest derivative of order α, l - 1 < α ≤ l . If β i < α are the other derivatives, the existing theory requires α - max ⁡ { β i } ≥ l - 1 for the existence of l linearly independent solutions. Thus, at most one derivative may have order greater than one, but all other derivatives must be between zero and one. We remove this essential restriction and construct l linearly independent solutions. With this aim, we remodel the series approaches and elaborate the multi-sum fractional series method in order to obtain the existence and linear independence results. We consider both Riemann–Liouville or Caputo fractional derivatives. [ABSTRACT FROM AUTHOR]

In 2004, Thakur introduced a positive characteristic analogue of multizeta values. Later, in 2017, he mentioned the two colored variants which are positive characteristic analogues of colored multizeta values in his survey of multizeta values in positive characteristic. In this paper, we study one of those two variants. We establish their fundamental properties, that include their non-vanishing, sum-shuffle relations, 푡-motivic interpretation and linear independence. For the linear independence results, we prove that there are no nontrivial k ̄ \overline{k} -linear relations among the colored multizeta values with different weights. [ABSTRACT FROM AUTHOR]

Let (t (m)) m ≥ 0 be Thue-Morse sequence and b > 2 be an integer. In this paper, we prove that the real numbers 1 , ∑ m = 0 ∞ t (m 2) b m + 1 and ∑ m = 0 ∞ t (m 3) b m + 1 are linearly independent over Q . [ABSTRACT FROM AUTHOR]

Topology identification of multilink graphs has received a lot of attention in recent years, however there have been no studies on topology identification of multilink hypergraphs. In this paper, therefore, the topology identification of multilink hypergraphs is investigated via the adaptive external synchronization method. A novel and easily verifiable linear independence condition is proposed with rigorous mathematical proof. By applying the linear independence condition, the sufficient condition for successfully realizing topology identification of multilink hypergraphs is obtained. Finally, the theoretical result is verified by numerical simulations. [ABSTRACT FROM AUTHOR]

Let (i , j) ∈ N × N ≥ 2 and S i , j be an infinite subset of positive integers including all prime numbers in some arithmetic progression. In this paper, we prove the linear independence over Q of the numbers 1 , ∑ n ∈ S i , j a i , j (n) b i n j , (i , j) ∈ N × N ≥ 2 , where b ≥ 2 is an integer and a i , j (n) are bounded nonzero integer-valued functions on S i , j . Moreover, we also establish a necessary and sufficient condition on the subset A of N × N ≥ 2 for the numbers 1 , ∑ n ∈ T i , j a i , j (n) b i n j , (i , j) ∈ A , to be linearly independent over Q for any given infinite subsets T i , j of positive integers. Our theorems generalize a result of V. Kumar. [ABSTRACT FROM AUTHOR]

Dual quaternions can represent rigid body motion in 3D spaces, and have found wide applications in robotics, 3D motion modelling and control, and computer graphics. In this paper, we introduce three different right linear independency concepts for a set of dual quaternion vectors, and study some related basic properties for dual quaternion vectors and dual quaternion matrices. We present a minimax principle for eigenvalues of dual quaternion Hermitian matrices. Based upon a newly established Cauchy-Schwarz inequality for dual quaternion vectors and singular value decomposition of dual quaternion matrices, we propose an inequality for singular values of dual quaternion matrices. Finally, we introduce the concept of generalized inverses of dual quaternion matrices, and present necessary and sufficient conditions for a dual quaternion matrix to be one of four types of generalized inverses of another dual quaternion matrix. [ABSTRACT FROM AUTHOR]

In [Appl. Comput. Harmon. Anal., 46 (2019), 664673] O. Christensen and M. Hasannasab observed that assuming the existence of an operator T sending en to en+1 for all n ∈ ℕ (where (en)n∈ℕ is a sequence of vectors) guarantees that (en)n∈ℕ is linearly independent if and only if dim{en}n∈ℕ = ∞. In this article, we recover this result as a particular case of a general order-theory-based model-theoretic result. We then return to the context of vector spaces to show that, if we want to use a condition like T(ei) = eϕ(i) for all i ∈ I where I is countable as a replacement of the previous one, the conclusion will only stay true if ϕ : I → I is conjugate to the successor function succ : n ↦n + 1 defined on ℕ. We finally prove a tentative generalization of the result, where we replace the condition T(ei) = eϕ(i) for all i ∈ I where ϕ is conjugate to the successor function with a more sophisticated one, and to which we have not managed to find a new application yet.

We investigate the linear dependence among odd periods of modular forms. We show that if the dimension of the space of cuspforms is greater than or equal to three, then any three odd periods are linearly independent. We also prove an asymptotic result for an arbitrary number of odd periods. The results are achieved by studying cuspidal projections of products of two Eisenstein series. [ABSTRACT FROM AUTHOR]

Commutation formulae with respect to a non-symmetric affine connection are obtained in this paper. The components of commutation formulae in this paper are covariant derivatives of tensors with respect to symmetric and non-symmetric affine connection. [ABSTRACT FROM AUTHOR]

This paper considers the topology identification issue for fractional complex networks (FCNs) based on the synchronization in finite-time. Firstly, two lemmas with respect to the convergence of a continuous differential function and its Caputo fractional derivative are developed, respectively. Secondly, a fractional auxiliary network, which consists of the isolated chaotic system, is introduced to avoid the requirement for the linear independence condition (LIC). A control protocol with the event-trigger mechanism (ETM) and a adaptive topology observer are designed to achieve the synchronization in finite time and topology identification. In addition, the synchronization conditions in finite time between regulated network and auxiliary network are addressed, and the topology identification is realized under the designed control protocol successfully. Finally, two numerical simulations are provided to illustrate the effectiveness of the proposed scheme and the validity of theoretical results. [ABSTRACT FROM AUTHOR]

This study investigated how undergraduate in-service mathematics teachers understand linear algebra concepts. The significance of the study is to reveal the difficulties and type of mental constructions they make when they are solving linear independence concepts by using inspection. The action, process, object, schema (APOS) theory was used to analyse the data collected. A descriptive qualitative case study design was used. The participants were 73 in-service teachers studying a Bachelor of Science Education Honours Degree in mathematics. Data was collected through a structured activity sheet and semi-structured interviews. The results showed that the teachers had an inadequate understanding of linear independence concepts. Many of the students developed the action conceptions of linear algebra as they were engrossed in step-by-step procedures trying to show that given vectors are linearly independent. The findings indicated that the schemata for determinants and the various theorems for determining linear independence promote learning. The modified genetic decomposition proposed has implications for teaching the concept and students should be given different kinds of sets to manipulate so that they can develop their mental constructions at the process level. [ABSTRACT FROM AUTHOR]

This paper introduces an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L2 function. Firstly, background and motivation for the conjecture are provided. Secondly, the main progress of this linear independence in the past twenty five years is reviewed. Finally, the partial results of the high dimensional case and other cases for the conjecture are briefly presented. [ABSTRACT FROM AUTHOR]

In this article, we prove the transcendence of special values of some Hurwitz zeta type series. Moreover, we find a linear independence criterion of these series under some mild conditions. We also show that, for any positive integer k and for any a , b ∈ (0 , 1) ∩ Q with a + b = 1 , at least one of the ζ (2 k , a) or ζ (2 k , b) must be transcendental. [ABSTRACT FROM AUTHOR]

We investigated understanding of the linear independence concept based on the type and nature of connections displayed in seven non-mathematics majors' interview responses to a set of open-ended questions. Through a qualitative analysis, we identified six categories of frequently displayed connections. There were also recognizable differences in the way the connections were applied by the participants. Overall, our findings pointed to an understanding in the form of two main clusters of connections. The two clusters were connected only by linear combination ideas. Each cluster, furthermore, was distinguishable via representation types. The first cluster contained arithmetic/algebraic modes and the second cluster included, mostly, geometric ideas. This paper discusses similarities and differences within and between clusters supported by participant responses. In light of the findings, we provide suggestions for the improvement of linear algebra education of non-mathematics student population. [ABSTRACT FROM AUTHOR]

We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values of analytic functions, how to prove that the Q -vector space spanned by 1 and those three numbers has dimension at least 3, whenever we are unable to achieve full linear independence, by using simultaneous approximations, i.e. those usually arising from Hermite–Padé approximations of type II and their suitable generalizations. It should be recalled that approximations of type I and II are related, at least in principle: when the numerical application consists in specializing actual functional constructions of the two types, they can be obtained, one from the other, as explained in a well-known paper by Mahler (1968) Compos Math 19: 95–166. That relation is reflected in a relation between the asymptotic behavior of the approximations at the infinite place of Q . Rather interestingly, the two view-points split away regarding the asymptotic behaviors at finite places (i.e. primes) of Q , and this makes the use of type II more convenient for particular purposes. In addition, sometimes we know type II approximations to a given set of functions, for which type I approximations are not known explicitly. Our approach can be regarded as a dual version of the standard linear independence criterion, which essentially goes back to Siegel. [ABSTRACT FROM AUTHOR]

Let M =(E, I) be a matroid and let S = {S1, . . . , St} be a family of subsets of E of size p. A subfamily ŝ ⊆ S is q-representative for S if for every set Y ⊆ E of size at most q, if there is a set X ∊ S disjoint from Y with X ∪ Y ∊I, then there is a set X ∪ Ŝ disjoint from Y with X ∪ Y ∊ I. By the classic result of Bollob ás, in a uniform matroid, every family of sets of size p has a q-representative family with at most ( p+q p ) sets. In his famous "two families theorem" from 1977, Lovász proved that the same bound also holds for any matroid representable over a field F. We give an efficient construction of a q-representative family of size at most ( p+q p ) in time bounded by a polynomial in ( p+q p ), t, and the time required for field operations. We demonstrate how the efficient construction of representative families can be a powerful tool for designing single-exponential parameterized and exact exponential time algorithms. The applications of our approach include the following: -In the LONG DIRECTED CYCLE problem, the input is a directed n-vertex graph G and the positive integer k. The task is to find a directed cycle of length at least k in G, if such a cycle exists. As a consequence of our 6.75k+o(k)nO(1) time algorithm, we have that a directed cycle of length at least log n, if such a cycle exists, can be found in polynomial time. -In the MINIMUM EQUIVALENT GRAPH (MEG) problem, we are seeking a spanning subdigraph D' of a given n-vertex digraph D with as few arcs as possible in which the reachability relation is the same as in the original digraph D. -We provide an alternative proof of the recent results for algorithms on graphs of bounded treewidth showing that many "connectivity" problems such as HAMILTONIAN CYCLE or STEINER TREE can be solved in time 2O(t)n on n-vertex graphs of treewidth at most t. For the special case of uniform matroids on nelements, we give a faster algorithm to compute a representative family. We use this algorithm to provide the fastest known deterministic parameterized algorithms for k- PATH, k-TREE, and, more generally, k-SUBGRAPH ISOMORPHISM, where the k-vertex pattern graph is of constant treewidth. [ABSTRACT FROM AUTHOR]

We introduce alternating multizeta values in positive characteristic which are generalizations of Thakur multizeta values. We establish their fundamental properties including non-vanishing, sum-shuffle relations, period interpretation and linear independence which is a direct sum result for these values. [ABSTRACT FROM AUTHOR]

Let $F_{n}$ and $L_{n}$ be the Fibonacci and Lucas numbers, respectively. Four corresponding zeta functions in $s$ are defined by $$\begin{eqnarray}\unicode[STIX]{x1D701}_{F}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{F}^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{F_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{L_{n}^{s}}},\quad \unicode[STIX]{x1D701}_{L}^{\ast }(s):=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{(-1)^{n+1}}{L_{n}^{s}}}.\end{eqnarray}$$ As a consequence of Nesterenko's proof of the algebraic independence of the three Ramanujan functions $R(\unicode[STIX]{x1D70C}),Q(\unicode[STIX]{x1D70C}),$ and $P(\unicode[STIX]{x1D70C})$ for any algebraic number $\unicode[STIX]{x1D70C}$ with $0 , the algebraic independence or dependence of various sets of these numbers is already known for positive even integers $s$. In this paper, we investigate linear forms in the above zeta functions and determine the dimension of linear spaces spanned by such linear forms. In particular, it is established that for any positive integer $m$ the solutions of $$\begin{eqnarray}\mathop{\sum }_{s=1}^{m}(t_{s}\unicode[STIX]{x1D701}_{F}(2s)+u_{s}\unicode[STIX]{x1D701}_{F}^{\ast }(2s)+v_{s}\unicode[STIX]{x1D701}_{L}(2s)+w_{s}\unicode[STIX]{x1D701}_{L}^{\ast }(2s))=0\end{eqnarray}$$ with $t_{s},u_{s},v_{s},w_{s}\in \mathbb{Q}$ $(1\leq s\leq m)$ form a $\mathbb{Q}$ -vector space of dimension $m$. This proves a conjecture from the Ph.D. thesis of Stein, who, in 2012, was inspired by the relation $-2\unicode[STIX]{x1D701}_{F}(2)+\unicode[STIX]{x1D701}_{F}^{\ast }(2)+5\unicode[STIX]{x1D701}_{L}^{\ast }(2)=0$. All the results are also true for zeta functions in $2s$ , where the Fibonacci and Lucas numbers are replaced by numbers from sequences satisfying a second-order recurrence formula. [ABSTRACT FROM AUTHOR]

The basis of T-splines are the point-based splines (PB splines) that are unstructured meshless splines. In this paper, we study associated PB splines with local knot vectors that are arbitrarily distributed in ([ 0 , 1 ] ∩ Q) d , d = 1 , 2 , where Q is the set of rational numbers. We prove the linear independence of linear PB splines under a mild assumption that their central knots are all distinct. The linearly independent property is one of important prerequisites for isogeometric analysis. Moreover, we illustrate that the same assumption can not be extended to two-dimensional case, by giving a set of linearly dependent bilinear PB splines. [ABSTRACT FROM AUTHOR]

In this paper we investigate arithmetic nature of the values of generalized hypergeometric functions and their derivatives. To solve the problem one often makes use of Siegel’s method. The first step in corresponding reasoning is, using the pigeonhole principle, to construct a functional linear approximating form, which has high order of zero at the origin of the coordinates.A hypergeometric function is defined as a sum of a power series whose coefficients are the products of the values of some rational function. Taken with the opposite sign, the zeroes of a numerator and a denominator of this rational function are called parameters of the corresponding generalized hypergeometric function. If the parameters are irrational it is impossible, as a rule, to employ Siegel’s method. In this case one applies the method based on the effective construction of the linear approximating form.Additional difficulties arise in case the rational function numerator involved in the formation of the coefficients of the hypergeometric function under consideration is different from the identical unit. In this situation even the availability of the effective construction of approximating form does not enable achieving an arithmetic result yet. In this paper we consider just such a case. To overcome difficulties arisen here we consider the values of the corresponding hypergeometric function and its derivatives at small points only and impose additional restrictions on parameters of the function.

To study an arithmetic nature of the values of hyper-geometric functions (and their derivatives including those with respect to parameter), it is common practice to use one either Siegel's method or the method based on the effective construction of the linear approximating form. The main distinction between these methods consists in the mode of construction of the first approximating form. Applying Siegel's method allows us to construct such a form by means of a pigeonhole principle that makes it possible to establish, in certain cases, the algebraic independence of the values of corresponding functions. The Siegel's method can be usually applied just while considering hyper-geometric functions with rational parameters. The effective method has a certain advantage here, for in some cases this method enables us to carry out corresponding investigation also for the functions with irrational parameters. Another advantage of the effective method is that it provides obtaining of high- accuracy quantitative results. By quantitative results one usually implies the estimates of the moduli of the linear forms in the values of corresponding functions. The effective method has made it possible to obtain, in some cases, estimates being precise regarding the height of such forms with calculation of the corresponding constants. A drawback of the effective method is the narrow domain of its applications. The effective construction of the linear approximating form, which initiates reasoning, is always a challenge. So far, an effective construction of the approximating form for the product of powers of hyper-geometric functions (with the rare exceptions) failed.In both aforementioned methods one proves previously linear independence (in Siegel's method, as a rule, algebraic independence,) of the functions under consideration. Such a proof is often considered as an independent result.In this paper, by means of the method especially elaborated for this purpose we establish linear independence of some hyper-geometric functions and their derivatives (including those with respect to parameter) over the field of rational fractions. Subsequently, it will be possible to apply this result to investigate arithmetic properties of the values of such functions. Herewith we mean the application of the effective method to achieve the sufficiently accurate quantitative result.

Topology identification of complex dynamical networks received extensive attention in the past decade. Most existing studies rely heavily on the linear independence condition (LIC). We find that a critical step in using this condition is not rigorous. Besides, it is difficult to verify this condition. Without regulating the original network, possible identification failure caused by network synchronization cannot be avoided. In this paper, we propose a new method to overcome these shortcomings. We add a regulation mechanism to the original network and construct an auxiliary network consisting of isolated nodes. Along with the outer synchronization between the regulated network and the auxiliary network, we show that the original network can be identified. Our method can avoid identification failure caused by network synchronization. Moreover, we show that there is no need to check the LIC. We finally provide some examples to demonstrate that our method is reliable and has good performances. [ABSTRACT FROM AUTHOR]

Linear independence of interval vectors is one of the most important issue in analyzing the controllability or observability of systems under uncertainty. From the viewpoint of quasilinear space, this paper aims to determine the linear independence of interval vectors by the conjunctive and disjunctive views of sets, respectively. We first show the concept of linear independence of interval vectors as conjunctive sets. And then, the dimension of ⟨ I R n , + , · ⟩ is investigated. Second, viewed the interval as a disjunctive set, we introduce the weak, strong, tolerable, and controllable linear independence of interval vectors, respectively. The method and algorithms are developed to check the weak, strong, tolerable, and controllable linear independence of a set of interval vectors. Moreover, the linear independence of fuzzy number vectors is studied, too. Finally, four numerical examples are provided to illustrate and substantiate our theoretical developments and established algorithms. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give linear independence results for the values of Lambert type series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences { n ℓ } ℓ ≥ 1 . For example, the three numbers 1 , ∑ p : prime 1 F p 2 , ∑ p : prime 1 L p 2 are linearly independent over Q (5) , where { F n } and { L n } are the Fibonacci and Lucas numbers, respectively. [ABSTRACT FROM AUTHOR]

Well-designed software systems, with providers only modules, have been rigorously obtained by algebraic procedures from the software Laplacian Matrices or their respective Modularity Matrices. However, a complete view of the whole software system should display, besides provider relationships, also consumer relationships. Consumers may have two different roles in a system: either internal or external to modules. Composite modules, including both providers and internal consumers, are obtained from the joint providers and consumers Laplacian matrix, by the same spectral method which obtained providers only modules. The composite modules are integrated into a whole Software System by algebraic connectors. These algebraic connectors are a minimal Occam's razor set of consumers external to composite modules, revealed through iterative splitting of the Laplacian matrix by Fiedler eigenvectors. The composite modules, of the respective standard Modularity Matrix for the whole software system, also obey linear independence of their constituent vectors, and display block-diagonality. The spectral method leading to composite modules and their algebraic connectors is illustrated by case studies. The essential novelty of this work resides in the minimal Occam's razor set of algebraic connectors — another facet of Brooks' Propriety principle leading to Conceptual Integrity of the whole Software System — within Linear Software Models, the unified algebraic theory of software modularity. [ABSTRACT FROM AUTHOR]

We study the Heil–Ramanathan–Topiwala conjecture in L p spaces by reformulating it as a fixed point problem. This reformulation shows that a function with linearly dependent time–frequency translates has a very rigid structure, which is encoded in a family of linear operators. This is used to give an elementary proof that if f ∈ L p (R) , p ∈ [ 1 , 2 ] , and Λ ⊆ R × R is contained in a lattice then the set of time frequency translates (f (a , b)) (a , b) ∈ Λ is linearly independent. Our proof also works for the case 2 < p < ∞ if Λ is contained in a lattice of the form α Z × β Z . [ABSTRACT FROM AUTHOR]

Most of existing research works in the field of feature selection via matrix factorization techniques have been employed for unsupervised learning problems. This paper introduces a new framework for the supervised feature selection, called supervised feature selection by constituting a basis for the original space of features and matrix factorization (SFS-BMF). To this end, SFS-BMF is a guided search to find a basis for the original space of features that inherently contains linearly independent features and can be replaced with the original space. For finding the best subset of features regarding the class attribute, information gain is utilized for the process of constructing a basis. In fact, a basis for the original features is constructed according to the most informative features in terms of the information gain. Then, this basis is decomposed through a matrix factorization form in order to select a subset of features. Our proposed method guarantees the maximum relevancy of selected features to the output by using the information gain while simultaneously secures the minimum redundancy among them based on the linear independence property. Several experiments on high-dimensional microarray datasets are conducted for illustrating the efficiency of SFS-BMF. The experimental results show that the proposed SFS-BMF method outperforms some state-of-the-art feature selection methods with respect to classification performance and also according to the computational complexity. [ABSTRACT FROM AUTHOR]

Let H n = ∑ k = 1 n 1 k be the nth harmonic number. Euler extended it to complex arguments and defined H r for any complex number r except for the negative integers. In this paper, we give a new proof of the transcendental nature of H r for rational r. For some special values of q > 1 , we give an upper bound for the number of linearly independent harmonic numbers H a / q with 1 ≤ a ≤ q over the field of algebraic numbers. Also, for any finite set of odd primes J with | J | = n , define W J = Q ¯ - span of { H 1 , H a j i / q i | 1 ≤ a j i ≤ q i - 1 , 1 ≤ j i ≤ q i - 1 , ∀ q i ∈ J }. Finally, we show that dim Q ¯ W J = ∑ i = 1 q i ∈ J n ϕ (q i) 2 + 2. [ABSTRACT FROM AUTHOR]

Reflection is an important part of teaching and needs to be considered carefully. In this study, we examined a mathematics instructor's reflections on teaching linear algebra. The research team employed Tall's (How humans learn to think mathematically: exploring the three worlds of mathematics. Cambridge University Press, Cambridge, 2013) framework to track the instructor's movements between the three worlds of mathematical thinking. The instructor emphasized the importance of symbolic and embodied thinking and made sure the students were ready before entering the formal world. It became evident that moving the class between the worlds appropriately at certain moments was critical to students' understanding of the concept, however at times required much anticipation and careful preparation in advance as well as creating appropriate teaching resources. The instructor's dual role as a researcher and participant resulted a model of instructional decision making which was used to give further insights during his movements between Tall's worlds and afforded him unique opportunities for reflection and subsequent carefully calculated class interventions. [ABSTRACT FROM AUTHOR]

In this study, I examined seven first-year linear algebra students' linear independence schemas. Data came from participants' interview responses to a set of nine questions. The analysis focused on the identification of concepts and connections pertaining to plans and activations. Overall, the findings revealed the existence of routinized plans, each containing one of six most frequently expressed connections. Moreover, I observed some participants activating these plans as fixed plans, and other participants activating them as ready-to-hand plans. Some activations were task specific. In short, I found participants' linear independence schemas to be populated by three main routinized plans, each with varying characteristics. [ABSTRACT FROM AUTHOR]