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In the evaluation scenario with undesirable outputs, the SBM-Undesirable model is usually used together with the SuperSBM-Undesirable model to fully differentiate efficient and inefficient decision-making units (DMUs). The existing SuperSBM-Undesirable model has two main forms, but both have problems in implementation. One form may be infeasible under the variable returns to scale (VRS); Another form has an inappropriate definition for undesirable output efficiency. These two problems are seriously ignored in the existing applications. In this paper, we first propose an improved SuperSBM-Undesirable model with strongly efficient projections, which is feasible under the constant returns to scale (CRS) or VRS technology. Then, by integrating the SBM-Undesirable and the improved SuperSBM-Undesirable models, we focus on proposing a concise, precise and practical one-stage model considering undesirable outputs to differentiate all DMUs and determine their strongly efficient projections simultaneously. The proposed one-stage model only contains essential decision variables and constraints, thereby effectively conserving computational time for large-scale practical applications. Further, by introducing the weighting preference of inputs and outputs, we construct the one-stage model with weighting preference to differentiate the importance of different indicators. Finally, through several data experiments, the superiority of our models in computational results and computational scale are verified.

This study is inspired by those of Huang et al. (Soft Comput. 25, 2513--2520, 2021) and Wang et al. (Inf. Sci. 179, 3026--3040, 2009) in which some ranking techniques for interval-valued intuitionistic fuzzy numbers (IVIFNs) were introduced. In this study, we prove that the space of all IVIFNs with the relation in the method for comparing any two IVIFNs based on a score function and three types of entropy functions is a complete chain and obtain that this relation is an admissible order. Moreover, we demonstrate that IVIFNs are complete chains to the relation in the comparison method for IVIFNs on the basis of score, accuracy, membership uncertainty index, and hesitation uncertainty index functions.

We prove that the space of intuitionistic fuzzy values (IFVs) with a linear order based on a score function and an accuracy function has the same algebraic structure as the one induced by a linear order based on a similarity function and an accuracy function. By introducing a new operator for IFVs via the linear order based on a score function and an accuracy function, we show that such an operator is a strong negation on IFVs. Moreover, we observe that the space of IFVs is a complete lattice and a Kleene algebra with the new operator. We also demonstrate that the topological space of IFVs with the order topology induced by the above two linear orders is not separable and metrizable but compact and connected. From some new perspectives,our results partially answer three open problems posed by Atanassov [Intuitionistic Fuzzy Sets: Theory and Applications, Springer, 1999] and [On Intuitionistic Fuzzy Sets Theory, Springer, 2012]. Furthermore, we construct an isomorphism between the spaces of IFVs and q-rung orthopedic fuzzy values (q-ROFVs) under the corresponding linear orders. To this end, we introduce the concept of admissible similarity measures with particular orders for IFSs, extending the existing definition of the similarity measure for IFSs, and construct an admissible similarity measure with a linear order based on a score function and an accuracy function, which is effectively applied to a pattern recognition problem about the classification of building materials.

We investigate the properties of the variable Lebesgue spaces with quasi-norm on a probability space, and give the atomic decompositions suited to the variable exponent martingale Hardy spaces. Using the decompositions and the harmonic mean of a variable exponent, we obtain several continuous embedding relations between martingale Hardy spaces with small exponent. Finally, we extend these results to the cases $0, 18 pages

In this article, it is proved that the maximal operator of one-dimensional dyadic derivative of dyadic integral I* and Cesaro mean operator σ* are bounded from the B-valued martingale Hardy spaces to Lα (0 < α < ∞), respectively. The facts show that it depends on the geometrical properties of the Banach space.

In this article, we study two types of martingale ergodic processes. We prove that a.e. convergence and L p convergence as well as maximal inequalities, which are established both in ergodic theory and martingale setting, also hold well for these new sequences of random variables. Moreover, the corresponding theorems in the former two areas turn out to be degenerate cases of the martingale ergodic theorems proved here.

In this article, the authors introduce some new Lorentz spaces for martingales, which are extensions of Hardy spaces of martingales. Then they discuss their basic properties, embedding relationships, and interpolation spaces between them, during which the use of rearrangement good-λ-inequality plays an important role.

In this article, the Banach space X and the martingales with values in it are considered. It is shown that the maximal operators of the one-dimensional dyadic derivative of the dyadic integral and Cesaro means are bounded from the dyadic Hardy- Lorentz space ra(X) to Lra(X) when X is isomorphic to a p-uniformly smooth space (1 < p ≤ 2). And it is also bounded from Hra(X) to Lra (X) (0 < r < ∞,0 < a ≤ ∞) when X has Radon-Nikodym property. In addition, some weak-type inequalities are given.

For Vilenkin-like system, the authors define a new operator H * f ≔ sup n | H n f |, where H n f is the weighted average for partial sums, and prove that H * is of type ( H* p ( G m ), L p ( G m )) for all ½ S* f ≔ | S n f | is of type (p,p) for 1 p > ∞, where S n f is the n -partial sum.

In this article, atomic decompositions and the duals of some B-valued r.v. sequence spaces are investigated. The results show that it closely depends on the geometrical properties of the sequence that take values in.

In this article, the authors introduce two operators—geometrical maximal operator M 0 and the closely related limiting operator M * 0 , then they give sufficient conditions under which the equality M 0 = M 0 holds, and characterize the equivalent relations between the weak or strong type weighted inequality and the property of W ∞ -weight or W * ∞ -weight for the geometrical maximal operator in the case of two-weight condition. What should be mentioned isthat the new operator—the geometrical minimal operator is equal to the limitation of the minimal operator sequence, and the results for the minimal operator proved in [12] makes the proof elegant and evident.

This paper is concerned with the composition operators between the area-type Nevanlinna classes. Some sufficient and necessary conditions are given in terms of the concept of Carleson measure and the standard techniques of Montel Theorem for the composition operator Cφ: Nap → Naq to be bounded or compact, where 1 < p ⩽ q. Moreover, the inducing maps which induce invertible or Fredholm composition operators on Nap are characterized.

The atomic decompositions of weak Hardy spaces of Banach-space-valued martingales are given. With the help of the atomic decompositions, some inequalities forB-valued martingales are established in the case 0

We establish a new characterization of AUMD (analytic unconditional martingale differences) spaces via biplurisubharmonic functions. That is,B∈AUMD iff there exists a bpsbh (biplurisubharmonic) functionL:B×B→[−∞, (x)) satisfyingL(x, 0),L(0,y)≥L(0,0)>0,L(x, y)≤L (0,0)+|x−y| andL(x, y)≤|x−y| for |x+y|+|x−y| ≥1. This provides an analogue of Piasecki's characterization of AUMD speces. Our arguments are based on some special properties of zigzag analytic martingales and martingale transforms.

The vector-valued Dirichlet-type spaces on the unit ball of Cn is introduced. We discuss the pointwise multipliers of Dirichlet-type spaces. Sufficient conditions of the pointwise multipliers of\(D_\mu ^2 for 0 \leqslant \mu< 2 if n = 1 or D_{\mu .q}^2 for 0< \mu< 1 if n \geqslant 2\) are given. Finally, Rademacherp-type space is characterized by vector-valued sequence spaces.

In martingale setting, it has been shown thatAp weights can be factorized in terms ofA1 weights. This factorization benefits many problems very much. In this paper, the new class of RH∞ plays the same role for RHs, which makes the reverse Holder inequalities hold with exponents>1, that the classA1 does forAp class. Therefore, the Jones' factorization theorem forAp weights was extended to include some information about the reverse Holder classes. And it is the most convenient object in weight theory indeed.

Let φ and ϕ be linear fractional self-maps of the unit diskD andX a separable Hilbert space. In this paper we completely characterize the weak compactness of the product operators of a composition operationCφ with another one’s adjointCρ* on the vector-valued Bergman spaceB1 (X) for formsCφCφ* andCφ*Cφ.

In order to improve the safety of high-energy solid propellants, a study is carried out for the effects of damage on the combustion of the NEPE (Nitrate Ester Plasticized Polyether) propellant. The study includes: (1) to introduce damage into the propellants by means of a large-scale drop-weight apparatus; (2) to observe microstructural variations of the propellant with a scanning electron microscope (SEM) and then to characterize the damage with density measurements; (3) to investigate thermal decomposition; (4) to carry out closed-bomb tests. The NEPE propellant can be considered as a viscoelastic material. The matrices of damaged samples are severely degraded, but the particles are not. The results of the thermal decomposition and closed-bomb tests show that the microstructural damage in the propellant affects its decomposition and burn rate.

We study the spacesH p, Sp, Kp, Σ p and Open image in new window of martingales with measure du=ϕd∪, where ϕ is a function with Clifford algebra values and satisfiesb p + -condition. We proved they are equivalent and isomorphic to the corresponding spaces of martingales with real measure d∪, respectively (1≤p

We give a new characterization ofq-uniform PL-convexity of complex Banach space by using the existence of a kind of functions with two variables and then prove a sharp weak (1, 1)-type inequality for analytic martingales with values in the Banach space.

We study boundedness properties of the q-mean-square operator S(q) on E-valued analytic martingales, where E is a complex quasi-Banach space and 2 ≤ q < ∞. We establish that a.s. finiteness of S(q) for every bounded E-valued analytic martingale implies strong (p, p)-type estimates for S(q) and all p ∈ (0, ∞). Our results yield new characterizations (in terms of analytic and stochastic properties of the function S(q)) of the complex spaces E that admit an equivalent q-uniformly PL-convex quasi-norm. We also obtain a vectorvalued extension (and a characterization) of part of an observation due to Bourgain and Davis concerning the Lp-boundedness of the usual square-function on scalar-valued analytic martingales.

LetX be a complex quasi-Banach space and Φ:[0,∞)→[0,∞) an increasing convex function with Φ(0)=0, \(\mathop {\lim }\limits_{t - \infty } \emptyset (t) = \infty \) and Φ∈Δ2. ThenLΦ* (X) is a quasi-Banach space with continuous quasi-norm andLΦ* (X) has the ARNP if and only ifX does.

Observation of the growth and formation of microcrack process zone by laser speckle interferometry for three kinds of rock. The effect of crack width and grain size on the process zone feature are also studied. The relation between the load F and crack tip opening displacement is examined by the same method

Using a simple method, we generalize Marcinkiewicz interpolation theorem to operators on Orlicz space and apply it to several important theorems in harmonic analysis.

According to experimental observation, a new cutting model for the strain-hardening material which conforms well with practice is put forward. Based on this model, the relationship between the cutting bending moment and the shape of the β family slip lines in the primary deformation zone is analysed theoretically and then the naturally curling radius of the chip is calculated from the hodograph. The theoretical analysis shows that the cutting bending moment impels the chip to curl naturally and the friction at the tool-chip interface arrests the tendency of the chip to curl. At the end of this paper, the situation of verifying tests is described and the reason for natural curling of the chip is discussed. The results of verifying tests show that the theoretical calculation is basically in accordance with practice.