Search Results (14)

We establish a theory whose structure is based on a fixed variable and an algebraic inequality and which improves the well-known least squares theory. The mentioned fixed variable plays a basic role in creating such a theory. In this direction, some new concepts, such as p-covariances with respect to a fixed variable, p-correlation coefficients with respect to a fixed variable, and p-uncorrelatedness with respect to a fixed variable, are defined in order to establish least p-variance approximations. We then obtain a specific system, called the p-covariances linear system, and apply the p-uncorrelatedness condition on its elements to find a general representation for p-uncorrelated variables. Afterwards, we apply the concept of p-uncorrelatedness for continuous functions, particularly for polynomial sequences, and we find some new sequences, such as a generic two-parameter hypergeometric polynomial of the ${}_{4}F_3$ type that satisfies a p-uncorrelatedness property. In the sequel, we obtain an upper bound for 1-covariances, an improvement to the approximate solutions of over-determined systems and an improvement to the Bessel inequality and Parseval identity. Finally, we generalize the concept of least p-variance approximations based on several fixed orthogonal variables. Full article

Numerous studies on the number pi ($\pi$) explore its properties, including normality and applicability. This research, grounded in two hypotheses, proposes and proves a theorem that employs a Bernoulli experiment to demonstrate the high probability of encountering any finite bit string within a sequence of consecutive bits in the decimal part of $\pi$. This aligns with findings related to its normality. To support the hypotheses, we present experimental evidence about the equiprobable and independent properties of bits of $\pi$, analyzing their distribution, and measuring correlations between bit strings. Additionally, from a cryptographic perspective, we evaluate the chaotic properties of two images generated using bits of $\pi$. These properties are evaluated similarly to those of encrypted images, using measures of correlation and entropy, along with two hypothesis tests to confirm the uniform distribution of bits and the absence of periodic patterns. Unlike previous works that solely examine the presence of sequences, this study provides, as a corollary, a formula to calculate an upper bound N. This bound represents the length of the sequence from $\pi$ required to ensure the location of any n-bit string at least once, with an adjustable probability p that can be set arbitrarily close to one. To validate the formula, we identify sequences of up to $n=$ 40 consecutive zeros and ones within the first N bits of $\pi$. This work has potential applications in Cryptography that use the number $\pi$ for random sequence generation, offering insights into the number of bits of $\pi$ required to ensure good randomness properties. Full article

This paper deals with p-maxian problem on cycles with an upper bound on the distances of all facilities. We consider the case of $p=2$ and show that, in the worst case, the optimal solution contains at least one vertex of the underlying cycle, which helps to develop an efficient algorithm to solve the constrained 2-maxian problem. Based on this property, we develop a linear time algorithm for the constrained 2-maxian problem on a cycle. We also discuss the relations between the constrained and unconstrained 2-maxian problems on which the underlying graphs are cycles. Full article

Optical flow is the apparent motion of the brightness patterns in an image. The pyramidal form of the Lucas-Kanade ($\mathrm{LK}$) method is frequently used for its computation but experiments have shown that the method has deficiencies. Problems arise because of numerical issues in the least squares ($\mathrm{LS}$) problem $min{∥Ax-b∥}_2^2$, $A\in {\mathbb{R}}^{m\times 2}$ and $m\gg 2$, which must be solved many times. Numerical properties of the solution $x_0=A^{†}b$ = ${\left(A^{T}A\right)}^{-1}{A}^{T}b$ of the $\mathrm{LS}$ problem are considered and it is shown that the property $m\gg 2$ has implications for the error and stability of $x_0$. In particular, it can be assumed that b has components that lie in the column space (range) $\mathcal{R}\left(A\right)$ of A, and the space that is orthogonal to $\mathcal{R}\left(A\right)$, from which it follows that the upper bound of the condition number of $x_0$ is inversely proportional to $cos\theta$, where $\theta$ is the angle between b and its component that lies in $\mathcal{R}\left(A\right)$. It is shown that the maximum values of this condition number, other condition numbers and the errors in the solutions of the $\mathrm{LS}$ problems increase as the pyramid is descended from the top level (coarsest image) to the base (finest image), such that the optical flow computed at the base of the pyramid may be computationally unreliable. The extension of these results to the problem of total least squares is addressed by considering the stability of the optical flow vectors when there are errors in A and b. Examples of the computation of the optical flow demonstrate the theoretical results, and the implications of these results for extended forms of the $\mathrm{LK}$ method are discussed. Full article

This study presents an investigation of the mix proportion and mechanical properties of one-part alkali-activated geopolymer concrete (GPC). The procedure for determining the mix proportion of one-part alkali-activated GPC, which uses a solid alkali activator in crystal form, is proposed. The proposed procedure was applied to a series of mixed proportions of GPC with different amounts of solid crystalline alkali activator (AA), water (W), and superplasticizer (SP), using the ratio between them to the total amount of binder (B, fly ash, and granulated blast furnace slag) by weight in order to evaluate their effect on the workability and compressive strength of the GPC. The slump, compressive and tensile strength, and elastic modulus of the one-part alkali-activated GPC were tested in various ways. The test results showed that solid crystalline alkali activators, water, and superplasticizers have significant effects on both the workability and compressive strength of GPC. The amount of one-part alkali activator should not exceed 12.0% of the total binder amount by weight (AA/B = 0.12) in order not to lose the workability of GPC. The minimum W–B ratio should be at least 0.43 to ensure the workability of the sample when no superplasticizer is added. An amount of 2.5% can be considered as the upper bound when using superplasticizer-based polysilicate for GPC. In addition, the elastic modulus and various types of tensile strength values of the one-part alkali-activated GPC were evaluated and compared with that predicted from compressive strength using equations given by two common ACI and Eurocode2 codes for ordinary Portland cement (OPC) concrete. Modifications of the equations showing the relationships between splitting tensile strength and compressive strength, as well as between elastic modulus and compressive strength and the development of compressive strength under the time provided by ACI and Eurocode2 for OPC concrete, were also made for one-part alkali-activated GPC. Full article

Supergrid graphs are derived by computing stitch paths for computerized embroidery machines. In the past, we have studied the Hamiltonian-related properties of supergrid graphs and their subclasses of graphs. In this paper, we propose a generalized graph class for supergrid graphs called extended supergrid graphs. Extended supergrid graphs include grid graphs, supergrid graphs, diagonal supergrid graphs, and triangular supergrid graphs as subclasses of graphs. In this paper, we study the problems of domination and independent domination on extended supergrid graphs. A dominating set of a graph is the subset of vertices on it, such that every vertex of the graph is in this set or adjacent to at least a vertex of this set. If any two vertices in a dominating set are not adjacent, this is called an independent dominating set. Domination and independent domination problems find a dominating set and an independent dominating set with the least number of vertices on a graph, respectively. The domination and independent domination set problems on grid graphs are known to be NP-complete, meaning that these two problems on extended supergrid graphs are also NP-complete. However, the complexities of these two problems in other subclasses of graphs remain unknown. In this paper, we first prove that these two problems on diagonal supergrid graphs are NP-complete, then, by a simple extension, we prove that these two problems on supergrid graphs and triangular supergrid graphs are also NP-complete. In addition, these two problems on rectangular supergrid graphs are known to be linearly solvable; however, the complexities of these two problems on rectangular triangular-supergrid graphs remain unknown. This paper provides tight upper bounds on the sizes of the minimum dominating and independent dominating sets for rectangular triangular-supergrid graphs. Full article

In several applications, the assumption of normality is often violated in data with some level of skewness, so skewness affects the mean’s estimation. The class of skew–normal distributions is considered, given their flexibility for modeling data with asymmetry parameter. In this paper, we considered two location parameter ($\mu$) estimation methods in the skew–normal setting, where the coefficient of variation and the skewness parameter are known. Specifically, the least square estimator (LSE) and the best unbiased estimator (BUE) for $\mu$ are considered. The properties for BUE (which dominates LSE) using classic theorems of information theory are explored, which provides a way to measure the uncertainty of location parameter estimations. Specifically, inequalities based on convexity property enable obtaining lower and upper bounds for differential entropy and Fisher information. Some simulations illustrate the behavior of differential entropy and Fisher information bounds. Full article

A bounded form of the Teissier distribution, namely the unit Teissier distribution, is introduced. It is subjected to a thorough examination of its important properties, including shape analysis of the main functions, analytical expression for moments based on upper incomplete gamma function, incomplete moments, probability-weighted moments, and quantile function. The uncertainty measures Shannon entropy and extropy are also performed. The maximum likelihood estimation, least square estimation, weighted least square estimation, and Bayesian estimation methods are used to estimate the parameters of the model, and their respective performances are assessed via a simulation study. Finally, the competency of the proposed model is illustrated by using two data sets from diverse fields. Full article

In recent years, complex-valued fuzzy metric spaces (in short CVFMS) were introduced by Shukla et al. (Fixed Point Theory 32 (2018)). This setting is a valuable extension of fuzzy metric spaces with the complex grade of membership function. They also established fixed-point results under contractive condition in the aforementioned spaces and generalized some essential existence results in fixed-point theory. The purpose of this manuscript is to derive some fixed-point results for multivalued mappings enjoying the least upper bound property in CVFMS. Furthermore, we studied the existence theorem for a unique solution to the Fuzzy fractional Volterra–Fredholm integro-differential equations (FCFVFIDEs) as an application to our derived result involving the Caputo derivative. Full article

A double Roman dominating function on a graph $G=(V,E)$ is a function $f:V\to \{0,1,2,3\}$ with the properties that if $f\left(u\right)=0$, then vertex u is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and if $f\left(u\right)=1$, then vertex u is adjacent to at least one vertex assigned 2 or 3. The weight of f equals $w\left(f\right)={\sum }_{v\in V}f\left(v\right)$. The double Roman domination number ${\gamma }_{dR}\left(G\right)$ of a graph G is the minimum weight of a double Roman dominating function of G. A graph is said to be double Roman if ${\gamma }_{dR}\left(G\right)=3\gamma \left(G\right)$, where $\gamma \left(G\right)$ is the domination number of G. We obtain the sharp lower bound of the double Roman domination number of generalized Petersen graphs $P(3k,k)$, and we construct solutions providing the upper bounds, which gives exact values of the double Roman domination number for all generalized Petersen graphs $P(3k,k)$. This implies that $P(3k,k)$ is a double Roman graph if and only if either $k\equiv 0$ (mod 3) or $k\in \{1,4\}$. Full article

The key to the asynchronous traffic shaping (ATS) technology being standardized in IEEE 802.1 time sensitive network (TSN) task group (TG) is the theorem that a minimal interleaved regulator (IR), attached to a FIFO system does not increase delay upper bound while suppresses the burst accumulation. In this work it is observed that the FIFO system can be a network for flows that share same input/output ports and same queues of the network, and are treated with a scheduling scheme that guarantees the FIFO property within a queue. Based on this observation, a framework for delay bound guarantee is further proposed, in which the networks with flow aggregates (FAs) scheduling and minimal IRs per FA attached at the network edge are interconnected. The framework guarantees the end-to-end delay bound with reduced complexity, compared to the traditional flow-based approach. Numerical analysis shows that the framework yields smaller bound than both the flow-based frameworks such as the integrated services (IntServ) and the class-based ATS, at least in the networks with identical flows and symmetrical topology. Full article

Starting with a new formulation for the mutual information (MI) between a pair of events, this paper derives alternative upper bounds and extends those to the case of two discrete random variables. Normalized mutual information (NMI) measures are then obtained from those bounds, emphasizing the use of least upper bounds. Conditional NMI measures are also derived for three different events and three different random variables. Since the MI formulation for a pair of events is always nonnegative, it can properly be extended to include weighted MI and NMI measures for pairs of events or for random variables that are analogous to the well-known weighted entropy. This weighted MI is generalized to the case of continuous random variables. Such weighted measures have the advantage over previously proposed measures of always being nonnegative. A simple transformation is derived for the NMI, such that the transformed measures have the value-validity property necessary for making various appropriate comparisons between values of those measures. A numerical example is provided. Full article

The number of applications of stimulus-responsive polymers is growing at an impressive rate. The motivation of this contribution is to use a commercially available low-budget silver-coated polyamide (PA6) as a thermo-responsive metal-polymer hybrid soft actuator. Polyamide is a hygroscopic polymer; therefore, its mechanical and physical-chemical properties are affected by exposition to humidity or immersion in water. The effect of water absorption content on the PA6 and silver-coated PA6 monofilament properties, such as mass change and resistance, were evaluated. Moreover, the influence of swelling and shrinking effects on the surface morphology, caused by variations of moisture and water immersion, was investigated. Based on these variations, the dynamics of the resistance of the hybrid material were analyzed in the context of the proposed hysteresis model. An identification procedure of the hysteresis is presented along with an approximation of the upper and lower bound based on a constrained least square approach. A switching logic algorithm for this hybrid dynamic system is introduced, which makes it possible to structure the non-linear function in a switching mode. Finally, a non-linear integral sliding manifold is proposed and tested to control the resulting force of the actuator.hysteresis model. An identification procedure of the hysteresis is presented along with an approximation of the upper and lower bound based on a constrained least square approach. A switching logic algorithm for this hybrid dynamic system is introduced, which makes it possible to structure the non-linear function in a switching mode. Finally, a non-linear integral sliding manifold is proposed and tested to control the resulting force of the actuator. Full article

A new kind of entropy will be introduced which generalizes both the differential entropy and the cumulative (residual) entropy. The generalization is twofold. First, we simultaneously define the entropy for cumulative distribution functions (cdfs) and survivor functions (sfs), instead of defining it separately for densities, cdfs, or sfs. Secondly, we consider a general “entropy generating function” φ, the same way Burbea et al. (IEEE Trans. Inf. Theory 1982, 28, 489–495) and Liese et al. (Convex Statistical Distances; Teubner-Verlag, 1987) did in the context of φ-divergences. Combining the ideas of φ-entropy and cumulative entropy leads to the new “cumulative paired φ-entropy” ( $CP{E}_{\phi }$ ). This new entropy has already been discussed in at least four scientific disciplines, be it with certain modifications or simplifications. In the fuzzy set theory, for example, cumulative paired φ-entropies were defined for membership functions, whereas in uncertainty and reliability theories some variations of $CP{E}_{\phi }$ were recently considered as measures of information. With a single exception, the discussions in the scientific disciplines appear to be held independently of each other. We consider $CP{E}_{\phi }$ for continuous cdfs and show that $CP{E}_{\phi }$ is rather a measure of dispersion than a measure of information. In the first place, this will be demonstrated by deriving an upper bound which is determined by the standard deviation and by solving the maximum entropy problem under the restriction of a fixed variance. Next, this paper specifically shows that $CP{E}_{\phi }$ satisfies the axioms of a dispersion measure. The corresponding dispersion functional can easily be estimated by an L-estimator, containing all its known asymptotic properties. $CP{E}_{\phi }$ is the basis for several related concepts like mutual φ-information, φ-correlation, and φ-regression, which generalize Gini correlation and Gini regression. In addition, linear rank tests for scale that are based on the new entropy have been developed. We show that almost all known linear rank tests are special cases, and we introduce certain new tests. Moreover, formulas for different distributions and entropy calculations are presented for $CP{E}_{\phi }$ if the cdf is available in a closed form. Full article