In this work, we introduce a analytical framework for proving the existence and uniqueness of solutions to nonlinear fractional differential equations (NFDEs) of order $ \nu \in (1, 2] $, subject to irregular boundary conditions, in a Banach space. Several fundamental definitions, theorems, and auxiliary lemmas are presented for the analysis of the proposed problem. We establish sufficient conditions for existence and uniqueness for the proposed system by applying Krasnosel'skii's fixed-point theorem and the Banach contraction principle. Finally, an illustrative example is constructed to validate the main theoretical results and demonstrate the effectiveness and applicability of the proposed approach, showing how the generalized framework can be systematically applied to analyze fractional boundary value problems.

In this study, the Jensen inequality has been refined. To achieve this, we employ the fact that an abstract convex function with certain structural properties can be bounded from below in a neighborhood of its global minimum in terms of the norm of its gradient. Some examples have been provided to show the extent of refinement.

Interconnection networks are commonly modeled as graphs, where vertices represent processors or devices and edges represent communication links. In such networks, locating or identifying components using a small set of reference points is fundamental for routing, fault diagnosis, monitoring, and navigation. A standard graph-theoretic measure of this capability is the metric dimension (or locating number), defined as the minimum cardinality of a resolving set whose distance vectors uniquely distinguish all vertices. Since determining the metric dimension is NP-hard in general, exact values for structured network families are of both theoretical and practical interest. In this paper, we studied the metric dimension of the line graph of an optimal $ 2 $-fault-tolerant token ring network. The underlying network $ \mathbb{T}_{m}^{2} $ augments a simple ring with additional links to ensure robust connectivity under up to two link or node failures, while the line graph $ L(\mathbb{T}_{m}^{2}) $ represents the network at the link level. A lemma was established to prove the lower bound of the metric dimension via contradiction, while the upper bound was determined by explicitly constructing resolving sets. The analysis was conducted case by case according to the congruence of the network order modulo $ 4 $, which simplified verification of all representation vectors. Our results showed that the fault-tolerant links increase the metric dimension compared with ordinary token rings, highlighting the influence of additional links on network distinguishability and providing insights for the design of robust interconnection networks.

This paper developed an analytical framework for a space-fractional Black-Scholes model formulated with the extended Caputo fractional derivative. Fundamental operational properties of the extended Mellin integral transform, including shift rules, transform formulas for Caputo-type derivatives of orders $ 0 < \alpha\leq1 $ and $ 1 < \beta\leq2 $, and a convolution theorem, were established and used to treat scale-invariant fractional differential equations. By applying the extended Mellin transform to the governing Cauchy problem, we derived an explicit integral representation of the solution involving a gamma-function-based time-evolution multiplier. The validity of the representation was rigorously verified, and the classical Black-Scholes model with dividends was recovered as a special case. The model was applied to European put options, with numerical results validating the method and illustrating the impact of fractional dynamics. Calibration to SPY option market data demonstrates that the fractional parameters $ \alpha $ and $ \rho $ enhance flexibility in fitting observed option prices and capturing market-dependent scaling effects.

In 1875, Glaisher systematically found several interesting determinant expressions of numbers, including Bernoulli, Cauchy, and Euler numbers. In this paper, we identify several determinants that express Euler polynomials. Goy and Shattuck presented several determinantal expressions of some families of Toeplitz–Hessenberg matrices with Tribonacci number entries. However, a determinant expression of Tribonacci numbers has not been studied much. By using a similar form of determinants to Euler's, we also give some determinant representations of generalized Tribonacci numbers.

The study of Dedekind's $ \eta $-function and its identities plays a significant role in number theory and combinatorics. In this paper, we study level 8 $ \eta $-function identities and their applications to colored partitions. Some of these identities arise from algebraic transformations of known mock theta function expansions. By applying these identities, we deduce combinatorial correspondences between specific classes of colored partitions with prescribed color restrictions. Our work extends existing methods and offers a deeper understanding of the combinatorial properties of partitions, contributing to both theoretical advancements and practical applications in partition theory.

The rapid development of the Internet of Vehicles and the increasing demand for mobile content services have significantly increased network traffic, leading to congestion and delays. To address these challenges, Content-centric IoV has emerged by integrating IoV with Content-Centric Networking, enabling efficient mobile content delivery. However, CIoV still faces limitations in outage zones where roadside unit coverage is restricted, hindering content transmission. To overcome this issue, we propose a SAC-based Precaching Vehicle Selection scheme that dynamically selects optimal precaching vehicles and determines appropriate content sizes to facilitate content delivery in outage zones. SAC-PVS operates on a snapshot-based inference model, which is specifically designed to optimize precaching decisions at the exact moment of a request without relying on continuous time-series monitoring. It leverages the SAC algorithm to handle continuous action spaces, enabling precise determination of both the optimal caching vehicles and the corresponding content quantities. A hierarchical reward structure penalizes excessive caching and traffic waste while rewarding successful content delivery in outage zones. Simulation results demonstrate that SAC-PVS outperforms both a comparable machine learning approach and a non-machine-learning baseline by reducing content delivery latency and minimizing traffic waste under dynamic vehicular conditions. The proposed SAC-PVS scheme improves the Quality of Service for vehicle users and optimizes network resource utilization for content delivery, offering a scalable and efficient solution for next-generation CIoV content services.