Maximal Function Characterizations of Hardy Spaces on Rn with Pointwise Variable Anisotropy

In 2011, Dekel et al. developed highly geometric Hardy spaces Hp(Θ), for the full range 0<p≤1, which were constructed by continuous multi-level ellipsoid covers Θ of Rn with high anisotropy in the sense that the ellipsoids can rapidly change shape from point to point and from level to level. In this article, when the ellipsoids in Θ rapidly change shape from level to level, the authors further obtain some real-variable characterizations of Hp(Θ) in terms of the radial, the non-tangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaces of Bownik.

Limits, Sequences, and Sums

This chapter first introduces the concepts of real sequences, Cauchy sequences, limits, and cluster points. The chapter then discusses functions of a real variable, continuity concepts, convergence concepts for sequences of functions, summability and order of magnitude relations, inequalities for sums, regular variation, and limit concepts for arrays.

Expectations

The expectation is defined, applying integration concepts to probability measures. Leading examples are given and the different characterizations of the expectation of a function are compared. The Markov and Jensen inequalities are given and consideration of multivariate distributions then leads to the treatment of the Cauchy–Schwarz, Hölder, Liapunov, Minkowski, and Loève inequalities. The final section treats the calculus of random functions of a real variable.

A dynamically controlled anaerobic/aerobic granular sludge reactor efficiently treats brewery/bottling wastewater

This study investigated the application of a dynamic control strategy in an aerobic granular sludge (AGS) reactor treating real variable brewery/bottling wastewater. For 482 days, the anaerobic and aerobic reaction steps in a lab-scale AGS system were controlled dynamically. A pH-based control was used for the anaerobic step, and an oxygen uptake rate (OUR) based control for the aerobic step. Additionally, the effect of an elongated aerobic step, and the effect of the removal of the suspended solids from the influent, on AGS formation were also investigated. In comparison to a static operation, the dynamic operation resulted in similar reactor performance, related to effluent quality and the anaerobic dissolved organic carbon (DOC) uptake efficiency, while the organic loading rate was significantly higher. The removal of suspended solids from the influent by chemical coagulation with FeCl3 turned hybrid floccular-granular sludge into fully granular sludge. The granulation coincided with a significant increase in the abundance of the glycogen-accumulating Candidatus Competibacter and an increase in the content of gel-forming EPS to respectively around 14 and 30%. In conclusion, this study showed the successful application of a dynamic control strategy based on common and low-cost sensors for AGS treatment of industrial wastewater.

bel and Euler Summation Formulas for SBV(R)

The purpose of this paper is to show that the natural setting for various Abel and Euler-Maclaurin summation formulas is the class of special function of bounded variation. A function of one real variable is of bounded variation if its distributional derivative is a Radom measure. Such a function decomposes uniquely as sum of three components: the first one is a convergent series of piece-wise constant function, the second one is an absolutely continuous function and the last one is the so-called singular part, that is a continuous function whose derivative vanishes almost everywhere. A function of bounded variation is special if its singular part vanishes identically. We generalize such space of special function of bounded variation to include higher order derivatives and prove that the functions of such spaces admit a Euler-Maclaurin summation formula. Such a result is obtained by deriving in this setting various integration by part formulas which generalizes various classical Abel summation formulas.

A New Class of Analytic Normalized Functions Structured by a Fractional Differential Operator

Newly, the field of fractional differential operators has engaged with many other fields in science, technology, and engineering studies. The class of fractional differential and integral operators is considered for a real variable. In this work, we have investigated the most applicable fractional differential operator called the Prabhakar fractional differential operator into a complex domain. We express the operator in observation of a class of normalized analytic functions. We deal with its geometric performance in the open unit disk.

Material Handling and Product Optimality of an Educational Institution Bakery Using Integer Programming

This study presents a numerical analysis for enhancing material handling allotment and deciding whether to produce or not in a bread production section of an educational institution. The study utilized 0/1 and real variables by the means of integer linear programming. In line with this, integer linear programming was applied in a bread production plant that produces three brands of bread, such as, medium size (x1), huge size (x2), and small size (x3). The discoveries using 0/1 variable shows that x1 = x2 = x3 = 1 while Z = 80 Naira. Hence, the item mix of bread is optimal, and production can continue. The result obtained utilizing real variable demonstrated that x1 (medium-size bread) = 649, x2 (huge-size bread) = 0, x3 (small-size bread) = 145, Z=18,388 Naira. In this manner the material handling allotment was optimized to obtain a profit of 18,388 Naira. Consequently, the organization should produce more medium-size bread (x1), more small-size bread (x3), and less huge-size bread (x2).

Serial encryption using the functions of real variable

Context. Using the functions of a real variable in cryptosystems as keys allow increasing their cryptographic strength since it is more difficult to select such keys. Therefore, the development of such systems is relevant. Objective. Cryptosystems with symmetric keys are proposed for encrypting and decrypting a sequence of characters represented as a one-dimensional numerical array of ASCII codes. These keys are functions of a real variable that satisfies certain restrictions. They can be both continuous and discrete. Method. Two cryptosystem options are proposed. In the first embodiment, the transmitting and receiving sides select two functions, the first transmitted symbol, the area of the function definition, and the step of changing the function argument. Discrete messages are encrypted by calculating the first-order integral disproportion of the encrypted array using a function. The corresponding value of the second function is added to the obtained cipher of each symbol for scrambling to complicate the analysis of the intercepted message. On the receiving side, the second function is subtracted and decryption performed by the inverse transformation of the formula for integral disproportion. In the second version, sequential encryption is performed when the cipher obtained using one of the key functions in the first stage is encrypted again by calculating the disproportion using the second function, the key. Accordingly, in two stages, decryption is performed. Results. Examples of encryption and decryption of a sequence of text characters are presented. It is shown that the same character is encoded differently depending on its position in the message. In the given examples it is presented the difficulty of key functions parameters choosing and the cryptographic strength of the proposed cryptosystem. Conclusions. Variants of the cryptosystem using the first-order integral disproportion function are proposed, in which the functions of a real variable serve as keys. To “crack” such a system, it is necessary not only to select the form of each function but also to find the values of its parameters with high accuracy. The system has high cryptographic strength.