An $L$-estimator is a form of linear function of order statistics. Xiang [8] investigated the bias of this estimator. In this investigation, we derive a formula for the mean square error of the smooth $L$-estimator.

In this paper, we address a fundamental question about the nature of bilinear operators first posed by Walter Rudin. While it has been known that there is not an open mapping theorem for bilinear operators in general, we show that bilinear operators enjoy an open mapping theorem, when the range has dimension three or less, and we address the simple, but crucial property that differentiates bilinear operators from their linear counterparts. We also give an example which shows that polynomials, in general do not enjoy an open mapping theorem, even when the range has dimension three.

We introduce the concepts of eKU/sKU/eUP/sUP-algebras as generalizations of the concepts of KU/UP-algebras, investigate some related properties of these new extensions of algebras, and show that the axioms of these new extensions are independent. We also introduce the concepts of eKU/sKU/eUP/sUP-subalgebras and eKU/sKU/eUP/sUP-filters of eKU/sKU/eUP/sUP-algebras, respectively, and describe the relationship between them. Moreover, we find the KU/UP-algebras induced by some congruence relations on eKU/eUP-algebras.

In this paper, we analyze properties of prime number sequences produced by the alternating sum of higher-order subsequences of the primes. We also introduce a new sieve which will generate these prime number sequences via the systematic selection and elimination of prime number indices on the real number line.

The purpose of this article is to introduce new types of limit points and separation axioms on supra topological spaces by using supra $\beta$-open sets. We explore some characterizations and explain the relationships between them with the help of examples. Also, we study many features of them and give sufficient conditions for some equivalent relations between them.

In this paper, we prove an index formula for $\text{Spin}^{\text{c}}$ Dirac operators on odd-dimensional $\mathbb{Z}/k\mathbb{Z}$-manifolds.

In this paper, we give definitions of generalized interval-valued fuzzy subsemigroups, which are generalizations of the concept of interval-valued fuzzy subsemigroups, and of $(\overline{\alpha}, \overline{\beta})$-interval-valued fuzzy subsemigroups, where $\overline{\alpha} \prec \overline{\beta}$. The relations between the generalized interval fuzzy subsemigroups and various subsemigroups are discussed. The main result is a characterization of weakly regular semigroups by using the properties of these generalized interval-valued fuzzy subsemigroups of semigroups.

Theoretical concepts of crisp graphs are highly utilized in computer science and applications. They are especially important in many research areas in computer science like image segmentation, data mining, clustering, network routing, and image capturing. If the role of vertices and edges are uncertain, having two opposite effects, positive and negative, then bipolar fuzzy graphs always play an important factor. In this paper, some important results on different types of operations of bipolar fuzzy graphs are improved. First, we explain some important theorems about the degree of composition, tensor product, and normal product of two bipolar fuzzy graphs using examples.

We present a simple criterion to determine if the graph of a rational function of the form $ f(x)=\frac{ax+b}{cx+d} $ includes a reflection of the reciprocal function $ r(x)=\frac{1}{x} $ over the $ x $-axis. In addition, we determine an algebraic expression for the factor by which the graph of $ y=r(x) $ is stretched vertically to produce the graph of $ y=f(x) $.