Search Results (35)

This paper examines temporal symmetry breaking and structural duality in a class of time-changed McKean–Vlasov neutral stochastic differential equations. The system features super-linear drift coefficients satisfying a one-sided local Lipschitz condition and incorporates a fundamental duality: one drift component evolves under a random time change $E_t$, while the other progresses in regular time t. Within the symmetric framework of mean-field interacting particle systems, where particles exhibit permutation invariance, we establish strong convergence of the tamed Euler–Maruyama method over finite time intervals. By replacing the one-sided local condition with a globally symmetric Lipschitz assumption, we derive an explicit convergence rate for the numerical scheme. Two numerical examples validate the theoretical results. Full article

In this study, we define the $k$-Cullen, $k$-Cullen–Lucas, and Modified $k$-Cullen sequences, and certain terms in these sequences are given. Then, we obtain the Binet formulas, generating functions, summation formulas, etc. In addition, we examine the relations among the terms of the $k$-Cullen, $k$-Cullen–Lucas, Modified $k$-Cullen, Cullen, Cullen–Lucas, Modified Cullen, $k$-Woodall, $k$-Woodall–Lucas, Modified $k$-Woodall, Woodall, Woodall–Lucas, and Modified Woodall sequences. The generating functions were derived and analyzed, especially for cases where Fibonacci numbers were assigned to parameter $k.$ Graphical representations of the generating functions and their logarithmic transformations revealed interesting growth trends and convergence behavior. Further, by multiplying the generating functions with exponential expressions such as $e^k$, we explored the self-similar nature and mirrored dynamics among the sequences. Specifically, it was observed that the Modified Cullen sequence exhibited a symmetric and inverse-like resemblance to the Cullen and Cullen–Lucas sequences, suggesting the presence of deeper structural dualities. Additionally, indefinite integrals of the generating functions were computed and visualized over a range of Fibonacci-indexed k values. These integral-based graphs further reinforced the phenomenon of symmetry and self-similarity, particularly in the Modified Cullen sequence. A key insight of this study is the discovery of a structural duality between the Modified Cullen and standard Cullen-type sequences, supported both algebraically and graphically. This duality suggests new avenues for analyzing generalized recursive sequences through generating function transformations. This observation provides new insight into the structural behavior of generalized Cullen-type sequences. Full article

In logical geometry, Aristotelian diagrams are studied in a systematic fashion. Recent developments in this field have shown that the square of opposition generalizes in two ways, which correspond precisely to the theory of opposition (leading to $\alpha$-structures) and the theory of implication (leading to ladders) it exhibits. These two kinds of Aristotelian diagrams are dual to each other, in the sense that they are the oppositional and implicative counterpart of the same construction. This paper formalizes this duality as $OI$-companionship, explores its properties, and applies it to various $\sigma$-diagrams. This investigation shows that $OI$-companionship has some interesting, but unusual behaviors. While it is symmetric, and works well on the level of Aristotelian families, it lacks (ir)reflexivity, transitivity, functionality, and seriality. However, we show that all important Aristotelian families from the literature do have a unique $OI$-companion. These findings explore the limits that arise when extending the duality between opposition and implication beyond the limits of $\alpha$-structures and ladders. Full article

For a given incomplete context, object-induced approximate concepts have been defined, and this type of approximate concept can induce a type of decision rule. Based on the duality principle, another set of approximate concepts can be defined from the perspective of attributes, i.e., attribute-induced approximate concepts. Although object induced approximate concepts and attribute induced approximate concepts are symmetrical by duality principle, their induced decision rules exhibit different properties and the connections between attribute induced decision rules and object induced decision rules in incomplete formal contexts are not clear. To this end, a type of attribute-induced approximate concept and a method of extracting attribute-induced decision rules are presented. More importantly, it is revealed that given a type of decision rules, there must be corresponding decision rules of the other type, and both of them can provide some useful information, but they are not equivalent to each other. In other words, each type of decision rule can provide some unique and irreplaceable information. Full article

In this paper, we provide a symmetric formula and a duality formula relating multiple zeta values and zeta-star values. We find that the summation ${\sum }_{a+b=r-1}{(-1)}^a{\zeta }^{★}(a+2,{\left\{2\right\}}^{p-1}){\zeta }^{★}({\left\{1\right\}}^{b+1},{\left\{2\right\}}^q)$ equals ${\zeta }^{★}({\left\{2\right\}}^p,{\left\{1\right\}}^r,{\left\{2\right\}}^q)+{(-1)}^{r+1}{\zeta }^{★}({\left\{2\right\}}^q,r+2,{\left\{2\right\}}^{p-1})$. With the help of this equation and Zagier’s ${\zeta }^{★}({\left\{2\right\}}^p,3,{\left\{2\right\}}^q)$ formula, we can easily determine ${\zeta }^{★}({\left\{2\right\}}^p,1,{\left\{2\right\}}^q)$ and several interesting expressions. Full article

In order to treat the degree of belief rationally, Baoding Liu created uncertainty theory. An uncertain variable, as a measurable function from an uncertainty space to the set of real numbers, is a basic concept in uncertainty theory. It is very meaningful to study its properties. Lusin’s theorem is one of the most classical theorems in measure theory that reveals the close relationship between measurable and continuous functions, and has important significance. In this paper, we give three pairs of continuity conditions for uncertain measures, and present that every pair reveals duality, which is a kind of symmetry between objects. Furthermore, it is demonstrated that these continuity conditions are equivalent. And, we also prove that these three pairs of continuity conditions and the condition: if $\left\{{\Lambda }_n\right\}$ is a sequence of open sets and ${\Lambda }_n\searrow \varnothing$, then ${\displaystyle \underset{n\to \infty }{lim}\mathcal{M}\left\{{\Lambda }_n\right\}=0}$ are equivalent in compact metric spaces. It is shown that Lusin’s theorem for uncertain variables holds if and only if the uncertain measure satisfies any of the above continuity conditions in a compact metric space. And, Lusin’s theorem can be applied to uncertain variables with symmetric or asymmetric distributions. Finally, we provide several examples to illustrate applications of Lusin’s theorem for uncertain variables. As far as we know, our results are new in uncertainty theory. Full article

In the 7th century BCE, the Kushite king Tanwetamani commissioned his “Dream Stela”, which was to be erected in the Amun Temple of Jebel Barkal. The lunette of the stela features a dualistic artistic motif whose composition, meaning, and significance are understudied despite their potential to illuminate important aspects of royal Kushite ideology. On the lunette, there are two back-to-back offering scenes that appear at first glance to be nearly symmetrical, but that closer inspection reveals to differ in subtle but significant ways. Analysis of the iconographic and textual features of the motif reveals its rhetorical function in this royal context. The two strikingly similar but meaningfully different offering scenes represented the two halves of a Kushite “Double Kingdom” that considered Kush and Egypt together as a complementary geographic dual, with Tanwetamani presiding as king of both. This “Mirrored Motif” encapsulated the duality present in the Kushite ideology of kingship during the Twenty-Fifth Dynasty, which allowed Tanwetamani to reconcile the present imperial expansion of Kush with the history of Egyptian activity in Nubia. The lunette of the Dream Stela is therefore political art that serves to advance the Kushite imperial agenda. Full article

The interaction between projects and servers has grown significantly in complexity; thus, applying parallel calculations increases dramatically. However, it should not be ignored that parallel processing gives rise to synchronization constraints and delays, generating penalty costs that may overshadow the savings obtained from parallel processing. Motivated by this trade-off, this study investigates two special and symmetric systems of split–join structures: (i) parallel structure and (ii) serial structure. In a parallel structure, the project arrives, splits into m parallel groups (subprojects), each comprising n subsequent stages, and ends after all groups are completed. In the serial structure, the project requires synchronization after each stage. Employing a numerical study, we investigates the time profile of the project by focusing on two types of delays: delay due to synchronization overhead (occurring due to the parallel structure), and delay due to overloaded servers (occurring due to the serial structure). In particular, the author studies the effect of the number of stages, the number of groups, and the utilization of the servers on the time profile and performance of the system. Further, this study shows the efficiency of lower and upper bounds for the mean sojourn time. The results show that the added time grows logarithmically with m (parallelism) and linearly with n (seriality) in both structures. However, comparing the two types of split–join structures shows that the synchronization overhead grows logarithmically undr both parallelism and seriality; this yields an unexpected duality property of the added time to the serial system. Full article

In this paper, we present meanings of K-$G_f$-bonvexity/K-$G_f$-pseudobonvexity and their generalization between the above-notice functions. We also construct various concrete non-trivial examples for existing these types of functions. We formulate K-$G_f$-Wolfe type multiobjective second-order symmetric duality model with cone objective as well as cone constraints and duality theorems have been established under these aforesaid conditions. Further, we have validates the weak duality theorem under those assumptions. Our results are more generalized than previous known results in the literature. Full article

In this article, we consider a semi-local ring $\mathbf{S}={\mathbb{F}}_q+u{\mathbb{F}}_q$, where $u^2=u$, $q=p^s$ and p is a prime number. We define a multiplication $yb=\beta \left(b\right)y+\gamma \left(b\right)$, where $\beta$ is an automorphism and $\gamma$ is a $\beta$-derivation on $\mathbf{S}$ so that $\mathbf{S}[y;\beta ,\gamma ]$ becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of $\mathbf{S}[y;\beta ,\gamma ]$ and obtain the most striking results that are better than previous findings. We also determine the structural properties of 1-generator skew cyclic and skew-quasi cyclic codes. Further, We demonstrate remarkable results of the above-mentioned codes over $\mathbf{S}$. Finally, we find the duality of skew cyclic and skew-quasi cyclic codes using a symmetric inner product. These codes are further generalized to double skew cyclic and skew quasi cyclic codes and a table of optimal codes is calculated by MAGMA software. Full article

In this paper, we explore the gravitational collapse of matter (dust) under the effect of zero-point length $l_0$. During the gravitational collapse, we neglect the backreaction effect of pre-Hawking radiation (in the sense that it is a small effect and cannot prevent the formation of an apparent horizon), then we recast the internal metric of a collapsing star as a closed FRW universe for any spherically symmetric case and, finally, we obtain the minimal value for the scale factor, meaning that the particles never hit the singularity. We argue that the object emerging at the end of the gravitational collapse can be interpreted as Planck stars (black hole core) hidden inside the event horizon of the black hole, with a radius proportional to ${(GM{l}_0^2/c^2)}^{1/3}$. Quite interestingly, we found the same result for the radius of the Planck star using a free-falling observer point of view. In addition, we point out a correspondence between the modified Friedmann’s equations in loop quantum gravity and the modified Friedmann’s equation in string T-duality. In the end, we discuss two possibilities regarding the final stage of the black hole. The first possibility is that we end up with Planck-size black hole remnants. The second possibility is that the inner core can be unstable and, due to the quantum tunneling effect, the spacetime can undergo a black-hole-to-white-hole transition (a bouncing Planck star). Full article

In this paper, we focus on a developable surface tangent to a timelike surface along a curve in Minkowski 3-space, which is called the osculating developable surface of the timelike surface along the curve. The ruling of the osculating developable surface is parallel to the osculating Darboux vector field. The main goal of this paper is to classify the singularities of the osculating developable surface. To this end, two new invariants of curves are defined to characterize these singularities. Meanwhile, we also research the singular properties of osculating developable surfaces near their lightlike points. Moreover, we give a relation between osculating Darboux vector fields and normal vector fields of timelike surfaces along curves from the viewpoint of Legendrian dualities. Finally, some examples with symmetrical structures are presented to illustrate the main results. Full article

In this paper, we make use of the Riemann–Liouville fractional q-integral operator to discuss the class $S_{q,\delta }^{*}\left(\alpha \right)$ of univalent functions for $\delta >0,\alpha \in \mathbb{C}-\left\{0\right\}$, and $0<\left|q\right|<1$. Then, we develop convolution results for the given class of univalent functions by utilizing a concept of the fractional q-difference operator. Moreover, we derive the normalized classes ${\mathcal{P}}_{\delta ,q}^{\zeta }(\beta ,\gamma )$ and ${\mathcal{P}}_{\delta ,q}\left(\beta \right)$ ($0<\left|q\right|<1$, $\delta \ge 0,0\le \beta \le 1,\zeta >0)$ of analytic functions on a unit disc and provide conditions for the parameters $q,\delta ,\zeta ,\beta$, and $\gamma$ so that ${\mathcal{P}}_{\delta ,q}^{\zeta }(\beta ,\gamma )\subset S_{q,\delta }^{*}\left(\alpha \right)$ and ${\mathcal{P}}_{\delta ,q}\left(\beta \right)\subset S_{q,\delta }^{*}\left(\alpha \right)$ for $\alpha \in \mathbb{C}-\left\{0\right\}$. Finally, we also propose an application to symmetric q-analogues and Ruscheweh’s duality theory. Full article

In order to describe human uncertainty more precisely, Baoding Liu established uncertainty theory. Thus far, uncertainty theory has been successfully applied to uncertain finance, uncertain programming, uncertain control, etc. It is well known that the limit theorems represented by law of large numbers (LLN), central limit theorem (CLT), and law of the iterated logarithm (LIL) play a critical role in probability theory. For uncertain variables, basic and important research is also to obtain the relevant limit theorems. However, up to now, there has been no research on these limit theorems for uncertain variables. The main results to emerge from this paper are a strong law of large numbers (SLLN), a weak law of large numbers (WLLN), a CLT, and an LIL for Bernoulli uncertain sequence. For studying these theorems, we first propose an assumption, which can be regarded as a generalization of the duality axiom for uncertain measure in the case that the uncertainty space can be finitely partitioned. Additionally, several new notions such as weakly dependent, Bernoulli uncertain sequence, and continuity from below or continuity from above of uncertain measure are introduced. As far as we know, this is the first study of the LLN, the CLT, and the LIL for uncertain variables. All the theorems proved in this paper can be applied to uncertain variables with symmetric or asymmetric distributions. In particular, the limit of uncertain variables is symmetric in (c) of the third theorem, and the asymptotic distribution of uncertain variables in the fifth theorem is symmetrical. Full article

Non-Hermitian (NH) quantum theory has been attracting increased research interest due to its featured properties, novel phenomena, and links to open and dissipative systems. Typical NH systems include PT-symmetric systems, pseudo-Hermitian systems, and their anti-symmetric counterparts. In this work, we generalize the pseudo-Hermitian systems to their complex counterparts, which we call pseudo-Hermitian-φ-symmetric systems. This complex extension adds an extra degree of freedom to the original symmetry. On the one hand, it enlarges the non-Hermitian class relevant to pseudo-Hermiticity. On the other hand, the conventional pseudo-Hermitian systems can be understood better as a subgroup of this wider class. The well-defined inner product and pseudo-inner product are still valid. Since quantum simulation provides a strong method to investigate NH systems, we mainly investigate how to simulate this novel system in a Hermitian system using the linear combination of unitaries in the scheme of duality quantum computing. We illustrate in detail how to simulate a general P-pseudo-Hermitian-$\phi$-symmetric two-level system. Duality quantum algorithms have been recently successfully applied to similar types of simulations, so we look forward to the implementation of available quantum devices. Full article