Algorithms for Weighted Independent Transversals and Strong Colouring

An independent transversal (IT) in a graph with a given vertex partition is an independent set consisting of one vertex in each partition class. Several sufficient conditions are known for the existence of an IT in a given graph and vertex partition, which have been used over the years to solve many combinatorial problems. Some of these IT existence theorems have algorithmic proofs, but there remains a gap between the best existential bounds and the bounds obtainable by efficient algorithms. Recently, Graf and Haxell (2018) described a new (deterministic) algorithm that asymptotically closes this gap, but there are limitations on its applicability. In this article, we develop a randomized algorithm that is much more widely applicable, and demonstrate its use by giving efficient algorithms for two problems concerning the strong chromatic number of graphs.

Survival strategies of SMEs amidst the COVID-19 pandemic: application of SEM and fsQCA

Purpose As a global pandemic, the COVID-19 crisis has profoundly affected the development of local firms, threatening the survival of small and medium enterprises (SMEs). This study aims to present an integrated framework by investigating the impact of strategic tools (i.e. firms’ capability of business agility, marketing operational efficiency, optimisation of innovation capability [OIC], managing employees’ satisfaction and rethinking customers’ experience) on the survival strategies of SMEs amidst the COVID-19 pandemic. Design/methodology/approach The current study used data from managers of SMEs and conducted an asymmetrical analysis (i.e. structural equation modelling [SEM]) to investigate the factors influencing the survival strategies of SMEs amidst the COVID-19 pandemic. This study also applied an asymmetrical approach (i.e. fuzzy sets qualitative comparative analysis-fsQCA) to explore the causal recipes and analysis of the necessary conditions to identify the factors required to achieve the expected outcome. Findings Results from SEM support all hypotheses. Results from fsQCA with the same data set show that firms’ business agility and OIC are necessary conditions for SMEs’ survival strategies. The result from fsQCA also reveals multiple sufficient conditions to succeed SMEs’ survival strategies amidst the COVID-19 pandemic. Practical implications Findings prescribe how SMEs adapt to this vulnerable business condition by applying the strategic tools and recipes suggested for survival. Originality/value This research applied an innovative analysis to reveal necessary and sufficient conditions that conventional methods such as SEM have limited power. This pioneering research in the context of the COVID-19 pandemic is considered novel in terms of the prescriptive strategic recipes offered to SMEs to adapt to and survive in the crisis caused by COVID-19.

Relative asymptotic equivalence of dynamic equations on time scales

This paper aims to study the relative equivalence of the solutions of the following dynamic equations $y^{\Delta }(t)=A(t)y(t)$ y Δ ( t ) = A ( t ) y ( t ) and $x^{\Delta }(t)=A(t)x(t)+f(t,x(t))$ x Δ ( t ) = A ( t ) x ( t ) + f ( t , x ( t ) ) in the sense that if $y(t)$ y ( t ) is a given solution of the unperturbed system, we provide sufficient conditions to prove that there exists a family of solutions $x(t)$ x ( t ) for the perturbed system such that $\Vert y(t)-x(t) \Vert =o( \Vert y(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ y ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ , and conversely, given a solution $x(t)$ x ( t ) of the perturbed system, we give sufficient conditions for the existence of a family of solutions $y(t)$ y ( t ) for the unperturbed system, and such that $\Vert y(t)-x(t) \Vert =o( \Vert x(t) \Vert )$ ∥ y ( t ) − x ( t ) ∥ = o ( ∥ x ( t ) ∥ ) , as $t\rightarrow \infty $ t → ∞ ; and in doing so, we have to extend Rodrigues inequality, the Lyapunov exponents, and the polynomial exponential trichotomy on time scales.

Tree Inference: Response Time and Other Measures in a Binary Multinomial Processing Tree, Representation and Uniqueness of Parameters

A Multinomial Processing Tree (MPT) is a directed tree with a probability associated with each arc and partitioned terminal vertices. We consider an additional parameter for each arc, a measure such as time. Each vertex represents a process. An arc descending from a vertex represents selection of a process outcome. A source vertex represents processing beginning with stimulus presentation and a terminal vertex represents a response. An experimental factor selectively influences a vertex if changing the factor level changes parameter values on arcs descending from that vertex and no others. Earlier work shows that if each of two factors selectively influences a different vertex in an arbitrary MPT it is equivalent to one of two simple MPTs. Which applies depends on whether the two selectively influenced vertices are ordered by the factors or not. A special case, the Standard Binary Tree for Ordered Processes, arises if the vertices are ordered and the factor selectively influencing the first vertex changes parameter values on only two arcs. We derive necessary and sufficient conditions, testable by bootstrapping, for this case. Parameter values are not unique. We give admissible transformations for them. We calculate degrees of freedom needed for goodness of fit tests.

Iterative learning fault-tolerant control for discrete-time nonlinear systems subject to stochastic actuator faults

This paper is concerned with an iterative learning fault-tolerant control strategy for discrete-time nonlinear systems where actuator faults arbitrarily occur. First, the stochastic faults occurring in multiplicative and additive manner are considered. Then, statistical behaviors of both faults-corrupted control signals from the actuator to the plant and faults-free ones from the iterative learning controller to the actuator are analyzed. Meanwhile, sufficient conditions of convergence for the proposed strategy are established by resorting to the time-weighted norm technique. Finally, two numerical examples are provided to illustrate the effectiveness and reliability of the proposed results. Both theoretical analysis and simulations indicate that the developed strategy is satisfactory in preserving decent tracking accuracy of the addressed systems subject to actuator faults.

Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in $R^{n}\backslash\{0\}$

We consider the following singular semilinear problem $$ \textstyle\begin{cases} \Delta u(x)+p(x)u^{\gamma }=0,\quad x\in D ~(\text{in the distributional sense}), \\ u>0,\quad \text{in }D, \\ \lim_{ \vert x \vert \rightarrow 0} \vert x \vert ^{n-2}u(x)=0, \\ \lim_{ \vert x \vert \rightarrow \infty }u(x)=0,\end{cases} $$ { Δ u ( x ) + p ( x ) u γ = 0 , x ∈ D ( in the distributional sense ) , u > 0 , in D , lim | x | → 0 | x | n − 2 u ( x ) = 0 , lim | x | → ∞ u ( x ) = 0 , where $\gamma <1$ γ < 1 , $D=\mathbb{R}^{n}\backslash \{0\}$ D = R n ∖ { 0 } ($n\geq 3$ n ≥ 3 ) and p is a positive continuous function in D, which may be singular at $x=0$ x = 0 . Under sufficient conditions for the weighted function $p(x)$ p ( x ) , we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.

Minimum Number of Colours to Avoid k-Term Monochromatic Arithmetic Progressions

By recalling van der Waerden theorem, there exists a least a positive integer w=w(k;r) such that for any n≥w, every r-colouring of [1,n] admits a monochromatic k-term arithmetic progression. Let k≥2 and rk(n) denote the minimum number of colour required so that there exists a rk(n)-colouring of [1,n] that avoids any monochromatic k-term arithmetic progression. In this paper, we give necessary and sufficient conditions for rk(n+1)=rk(n). We also show that rk(n)=2 for all k≤n≤2(k−1)2 and give an upper bound for rp(pm) for any prime p≥3 and integer m≥2.

Optimal location of resources and Steiner symmetry in a population dynamics model in heterogeneous environments

The subject of this paper is inspired by Cantrell and Cosner (1989) and Cosner, Cuccu and Porru (2013). Cantrell and Cosner (1989) investigate the dynamics of a population in heterogeneous environments by means of diffusive logistic equations. An important part of their study consists in finding sufficient conditions which guarantee the survival of the species. Mathematically, this task leads to the weighted eigenvalue problem \(-\Delta u =\lambda m u \) in a bounded smooth domain \(\Omega\subset \mathbb{R}^N\), \(N\geq 1\), under homogeneous Dirichlet boundary conditions, where \(\lambda \in \mathbb{R}\) and \(m\in L^\infty(\Omega)\). The domain \(\Omega\) represents the environment and \(m(x)\), called the local growth rate, says where the favourable and unfavourable habitats are located. Then, Cantrell and Cosner (1989) consider a class of weights \(m(x)\) corresponding to environments where the total sizes of favourable and unfavourable habitats are fixed, but their spatial arrangement is allowed to change; they determine the best choice among them for the population to survive. In our work we consider a sort of refinement of the result above. We write the weight \(m(x)\) as sum of two (or more) terms, i.e. \(m(x)=f_1(x)+f_2(x)\), where \(f_1(x)\) and \(f_2(x)\) represent the spatial densities of the two resources which contribute to form the local growth rate \(m(x)\). Then, we fix the total size of each resource allowing its spatial location to vary. As our first main result, we show that there exists an optimal choice of \(f_1(x)\) and \(f_2(x)\) and find the form of the optimizers. Our proof relies on some results in Cosner, Cuccu and Porru (2013) and on a new property (to our knowledge) about the classes of rearrangements of functions. Moreover, we show that if \(\Omega\) is Steiner symmetric, then the best arrangement of the resources inherits the same kind of symmetry. (Actually, this is proved in the more general context of the classes of rearrangements of measurable functions.

On the Weak Solutions of a Delay Composite Functional Integral Equation of Volterra-Stieltjes Type in Reflexive Banach Space

Differential and integral equations in reflexive Banach spaces have gained great attention and hve been investigated in many studies and monographs. Inspired by those, we study the existence of the solution to a delay functional integral equation of Volterra-Stieltjes type and its corresponding delay-functional integro-differential equation in reflexive Banach space E. Sufficient conditions for the uniqueness of the solutions are given. The continuous dependence of the solutions on the delay function, the initial data, and some others parameters are proved.

Practical Criteria for H-Tensors and Their Application

Identifying the positive definiteness of even-order real symmetric tensors is an important component in tensor analysis. H-tensors have been utilized in identifying the positive definiteness of this kind of tensor. Some new practical criteria for identifying H-tensors are given in the literature. As an application, several sufficient conditions of the positive definiteness for an even-order real symmetric tensor were obtained. Numerical examples are given to illustrate the effectiveness of the proposed method.