Investigating Distributional Chaos for Operators on Fréchet Spaces

In this paper, various notions of chaos for continuous linear operators on Fréchet spaces are investigated. It is shown that an operator is Li–Yorke chaotic if and only if it is mean Li–Yorke chaotic in a sequence whose upper density equals one; that an operator is mean Li–Yorke chaotic if and only if it admits a mean Li–Yorke pair, if and only if it is distributionally chaotic of type 2, if and only if it has an absolutely mean irregular vector. As a consequence, mean Li–Yorke chaos is not conjugacy invariant for continuous self-maps acting on complete metric spaces. Moreover, the existence of invariant scrambled sets (with respect to certain Furstenberg families) of a class of weighted shift operators is proved.

Extension of vector-valued functions and weak–strong principles for differentiable functions of finite order

In this paper we study the problem of extending functions with values in a locally convex Hausdorff space E over a field $$\mathbb {K}$$ K , which has weak extensions in a weighted Banach space $${\mathcal {F}}\nu (\Omega ,\mathbb {K})$$ F ν ( Ω , K ) of scalar-valued functions on a set $$\Omega$$ Ω , to functions in a vector-valued counterpart $$\mathcal {F}\nu (\Omega ,E)$$ F ν ( Ω , E ) of $${\mathcal {F}}\nu (\Omega ,\mathbb {K})$$ F ν ( Ω , K ) . Our findings rely on a description of vector-valued functions as continuous linear operators and extend results of Frerick, Jordá and Wengenroth. As an application we derive weak-strong principles for continuously partially differentiable functions of finite order and vector-valued versions of Blaschke’s convergence theorem for several spaces.

Associations Between First Parity Wean-to-Service Interval and Sow Lifetime Productivity Traits

This study aimed to 1) investigate associations between first parity wean-to-service interval (WSI) and sows’ lifetime reproductive traits and 2) identify cut-off values for WSI associated with lifetime traits. Data collected in 3,900-sows of farrow-to-finish commercial farm in Yucatan, Mexico. Lifetime productivity records including parity number at culling (NPC), lifetime number piglets born alive (LNBA) from parity two until culling, lifetime non-productive days (LNPD) and length of productive life (LPL) for sows were used. Association between WSI and sow productive traits were evaluated using general linear models, including year and season as categorical fixed effects and WSI as a continuous linear and quadratic predictors. Cut-off values for WSI were estimated using regression tree analysis. WSI was associated (P < 0.05) with LNBA (linear = -0.62 ± 0.025; quadratic 0.02 ± 0.008) and NCP (linear = -0.04 ± 0.018). Similarly, an association (P < 0.05) was observed between WSI and LNPD (linear = 2.81 ± 0.687; quadratic -0.05 ± 0.023). Cut-off values for WSI varied according to each of the predicted variables: WSI > 5 days would translate into longer 13 more days of LPL, WSI < 7 days would increase LNBA by two extra pigs, WSI ≥ 9 days increases NCP by 0.2 parities, and WSI < 10 days would mean 24 fewer LNPD. Shorter WSI during the first parity was associated with improved lifetime productivity traits. The estimated cut-off values for WSI could be used by producers, to decide when to implement strategies to improve management.

On derivatives of fuzzy multi-dimensional mappings and applications under generalized differentiability

This article identifies and presents the generalized difference (g-difference) of fuzzy numbers, Fréchet and Gâteaux generalized differentiability (g-differentiability) for fuzzy multi-dimensional mapping which consists of a new concept, fuzzy g-(continuous linear) function; Moreover, the relationship between Fréchet and Gâteaux g-differentiability is studied and shown. The concepts of directional and partial g-differentiability are further framed and the relationship of which will the aforementioned concepts are also explored. Furthermore, characterization is pointed out for Fréchet and Gâteaux g-differentiability; based on level-set and through differentiability of endpoints real-valued functions a characterization is also offered and explored for directional and partial g-differentiability. The sufficient condition for Fréchet and Gâteaux g-differentiability, directional and partial g-differentiability based on level-set and through employing level-wise gH-differentiability (LgH-differentiability) is expressed. Finally, to illustrate the ability and reliability of the aforementioned concepts we have solved some application examples.

Solving Poisson Equation by Distributional HK-Integral: Prospects and Limitations

In this paper, we present some properties of integrable distributions which are continuous linear functional on the space of test function D ℝ 2 . Here, it uses two-dimensional Henstock–Kurzweil integral. We discuss integrable distributional solution for Poisson’s equation in the upper half space ℝ + 3 with Dirichlet boundary condition.

On linear continuous operators on C_p-spaces

The paper describes the structure of a linear continuous operator on the space of continuous functions in the topology of pointwise convergence. The corresponding theorem is a generalization of A.V.Arkhangel'skii's theorem on the general form of a continuous linear functional on such spaces.

Oscillation analysis of solutions of non-zero continuous linear functional equations

As a mathematical model of mechanical and electronic oscillation, the study and analysis of the oscillation characteristics of the solution of the non-zero continuous linear functional equation are of great significance in theory and practice. In view of the oscillation characteristics of the solutions of the second and third order non-zero continuous functional equations, this paper puts forward a hypothesis, studies the oscillation and asymptotics of the non-zero continuous linear functional differential equations by using the generalized Riccati transformation and the integral average technique, and establishes some new sufficient conditions for the oscillation or convergence to zero of all solutions of the equations, so as to obtain a new theorem for the solutions of the non-zero continuous linear functional equations.