Search Results (62)

Financial document understanding remains a critical challenge for Large Language Models, primarily due to the complex interplay between narrative text and structured numerical tables. Existing Retrieval-Augmented Generation (RAG) systems often treat these modalities in isolation, leading to significant failures in tasks requiring joint reasoning. This study introduces HierFinRAG, a novel hierarchical multimodal framework designed to unify tabular and textual data processing. Our approach employs a Table-Text Graph Neural Network (TTGNN) to explicitly model semantic and structural dependencies between table cells and corresponding text, coupled with a Symbolic–Neural Fusion module that routes queries between a neural generator and a symbolic calculator for precise arithmetic operations. We evaluate the system on the FinQA and FinanceBench datasets, comparing performance against strong baselines including Vanilla RAG and GPT-4o with Code Interpreter. Results demonstrate that HierFinRAG achieves an Exact Match score of 82.5% on FinQA, surpassing the best baseline by 6.5 percentage points, while maintaining a 3.5× faster inference latency than agentic approaches. These findings indicate that integrating hierarchical structural awareness with hybrid reasoning significantly enhances the accuracy and interpretability of financial artificial intelligence systems. Full article

In this paper, new fast algorithms for computing the discrete cosine transform type VI (DCT-VI) are proposed, with a special emphasis on short input sequences of three to eight samples. Fast algorithms for small discrete trigonometric transformations are directly used for efficient processing of small data sets and also serve as fundamental building blocks for constructing algorithms for larger trigonometric transforms. By exploiting the intrinsic structural properties of the DCT-VI matrices of different sizes, the proposed methods significantly reduce arithmetic complexity compared to the conventional matrix–vector multiplication approach. The paper presents a detailed mathematical formulation of the algorithms, supported by data-flow graphs that illustrate the computational structure and facilitate the precise estimation of arithmetic operations. Optimized pseudocode implementations incorporating variable reuse are also introduced to facilitate practical realization in software environments. Performance analysis demonstrates a substantial reduction in the number of multiplications (up to 66%) and a slight decrease in additions (approximately 9%) for input sizes ranging from three to eight, thereby improving the execution speed of the considering transform. The proposed algorithms are well-suited for applications in video coding, data compression, and digital signal processing, where computational efficiency is critical. Full article

Let $\mathcal{Z}$ be a graph of order n with m edges. Let $A_{ag}\left(\mathcal{Z}\right)$ represents the arithmetic–geometric matrix of $\mathcal{Z}$. The eigenvalues of the matrix $A_{ag}\left(\mathcal{Z}\right)$ are called the arithmetic–geometric eigenvalues, and the eigenvalue with the largest modulus is called the arithmetic–geometric spectral radius of $\mathcal{Z}$. The sum of the absolute values of the arithmetic–geometric eigenvalues is called the arithmetic–geometric energy of $\mathcal{Z}$. In this paper, we establish sharp upper and lower bounds for the AM-GM spectral radius in terms of various graph parameters and provide a complete characterization of the extremal graphs that attain these bounds. Additionally, we derive new bounds for the AM-GM energy of a graph and identify the corresponding extremal structures. In both contexts, our results significantly improve upon several existing bounds reported in the literature. Full article

This paper presents new fast algorithms for the type VII discrete cosine transform (DCT-VII) applied to input data sequences of lengths ranging from 3 to 8. Fast algorithms for small-sized trigonometric transforms enable the processing of small data blocks in image and video coding with low computational complexity. To process the information in image and video coding standards, the fast DCT-VII algorithms can be used, taking into account the relationships between the DCT-VII and the type II discrete cosine transform (DCT-II). Additionally, such algorithms can be used in other digital signal processing tasks as components for constructing algorithms for large-sized transforms, leading to reduced system complexity. Existing fast odd DCT algorithms have been designed using relationships among discrete cosine transforms (DCTs), discrete sine transforms (DSTs), and the discrete Fourier transform (DFT); among different types of DCTs and DSTs; and between the coefficients of the transform matrix. However, these algorithms require a relatively large number of multiplications and additions. The process of obtaining such algorithms is difficult to understand and implement. To overcome these shortcomings, this paper applies a structural approach to develop new fast DCT-VII algorithms. The process begins by expressing the DCT-VII as a matrix-vector multiplication, then reshaping the block structure of the DCT-VII matrix to align with matrix patterns known from the basic papers in which the structural approach was introduced. If the matrix block structure does not match any known pattern, rows and columns are reordered, and sign changes are applied as needed. If this is insufficient, the matrix is decomposed into the sum of two or more matrices, each analyzed separately and transformed similarly if required. As a result, factorizations of DCT-VII matrices for different input sequence lengths are obtained. Based on these factorizations, fast DCT-VII algorithms with reduced arithmetic complexity are constructed and presented with pseudocode. To illustrate the computational flow of the resulting algorithms and their modular design, which is suitable for VLSI implementation, data-flow graphs are provided. The new DCT-VII algorithms reduce the number of multiplications by approximately 66% compared to direct matrix-vector multiplication, although the number of additions decreases by only about 6%. Full article

Cryptography stands out as a scientific methodology for safeguarding communication against unauthorized access. This article proposes a newly formulated graph termed the Index Difference Graph (IDG). The proposed graph model serves as the secret key in the encryption process. Furthermore, we present a new graph-based algorithm, the Index Difference Modular Cryptographic (IDMC) Algorithm, and analyze it using centipede and path graphs. The goal of this graph-based approach is to increase the encryption rate while maintaining computational efficiency. This research investigates different types of index difference graphs and analyzes the time and space complexity of the algorithm. IDMC exhibits a lower collision probability, thereby enhancing encryption security. When employing a graph that admits an Index Difference Graph structure in the cryptographic algorithm, both the sender and receiver must be aware of the graph’s precise structure, as this strengthens the robustness of the cryptographic key. The application of the index difference centipede graph $P_n\odot 2k_1$ in cryptography, examined through the IDMC algorithm, demonstrates exceptionally high brute-force resistance estimated at approximately $2.6\times {10}^{39}$ for smaller instances with $n\le 7$ and escalating to $6.93\times {10}^{163}$ for larger graphs with $n\ge 20$. This resistance underscores the algorithm’s efficiency and cryptographic resilience. Full article

The growing importance of social networks has led to increased research into trust estimation and interpretation among network entities. It is important to predict the trust score between users in order to minimize the risks in user interactions. This article enables the identification of the most reliable and least reliable entities in a network by expressing trust scores numerically. In this paper, the social network is modeled as a graph, and trust scores are calculated by taking the powers of the ratio matrix between entities and summing them. Taking the power of the proportion matrix based on the number of entities in the network requires a lot of arithmetic load. After taking the powers of the eigenvalues of the ratio matrix, these are multiplied by the eigenvector matrix to obtain the power of the ratio matrix. In this way, the arithmetic cost required for calculating trust between entities is reduced. This paper calculates the trust score between entities using linear algebra techniques to reduce the arithmetic load. Trust detection algorithms use shortest paths and similar methods to eliminate paths that are deemed unimportant, which makes the result questionable because of the loss of data. The novelty of this method is that it calculates the trust score without the need for explicit path numbering and without any data loss. Full article

In this work, we explore Wirtinger-type inequalities involving tempered Ψ-Caputo fractional derivatives by utilizing Taylor’s formula. We establish more general inequalities for the same operator in $L_p$ norms for $p>1$ by using Hölder’s inequality. Special cases are discussed in the form of remarks by highlighting their relationships with the existing literature. The derived results are also verified through illustrative examples, including tables and graphs. Moreover, applications of the obtained inequalities are discussed in the context of arithmetic and geometric means. Full article

This paper presents new fast algorithms for the fourth type discrete Hartley transform (DHT-IV) for input data sequences of lengths from 3 to 8. Fast algorithms for small-sized trigonometric transforms can be used as building blocks for synthesizing algorithms for large-sized transforms. Additionally, they can be utilized to process small data blocks in various digital signal processing applications, thereby reducing overall system latency and computational complexity. The existing polynomial algebraic approach and the approach based on decomposing the transform matrix into cyclic convolution submatrices involve rather complicated housekeeping and a large number of additions. To avoid the noted drawback, this paper uses a structural approach to synthesize new algorithms. The starting point for constructing fast algorithms was to represent DHT-IV as a matrix–vector product. The next step was to bring the block structure of the DHT-IV matrix to one of the matrix patterns following the structural approach. In this case, if the block structure of the DHT-IV matrix did not match one of the existing patterns, its rows and columns were reordered, and, if necessary, the signs of some entries were changed. If this did not help, the DHT-IV matrix was represented as the sum of two or more matrices, and then each matrix was analyzed separately, if necessary, subjecting the matrices obtained by decomposition to the above transformations. As a result, the factorizations of matrix components were obtained, which led to a reduction in the arithmetic complexity of the developed algorithms. To illustrate the space–time structures of computational processes described by the developed algorithms, their data flow graphs are presented, which, if necessary, can be directly mapped onto the VLSI structure. The obtained DHT-IV algorithms can reduce the number of multiplications by an average of 75% compared with the direct calculation of matrix–vector products. However, the number of additions has increased by an average of 4%. Full article

Recently, the exponential arithmetic–geometric index ($EAG$) was introduced. The exponential arithmetic–geometric index ($EAG$) of a graph G is defined as $EAG\left(G\right)={\displaystyle \sum _{v_i{v}_j\in E\left(G\right)}}{e}^{{\displaystyle \frac{d_i+d_j}{2\sqrt{d_i{d}_j}}}}$, where $d_i$ represents the degree of the vertex $v_i$ in G. The characterization of extreme structures in relation to graph invariants from the class of unicyclic graphs is an important problem in discrete mathematics. Cruz et al., 2022 proposed a unified method for finding extremal unicyclic graphs for exponential degree-based graph invariants. However, in the case of $EAG$, this method is insufficient for generating the maximal unicyclic graph. Consequently, the same article presented an open problem for the investigation of the maximal unicyclic graph with respect to this invariant. This article completely characterizes the maximal unicyclic graph in relation to $EAG$. Full article

In the field of neuroeconomics, the assessment of cognitive workload is a crucial issue with significant implications for real-world applications. Previous research has made progress in task-based germane cognitive load classification, but decentralized studies focusing on task-independent assessment have often produced less than optimal results. In this study, we present a stacked graph attention convolutional networks (SGATCNs) model to tackle the challenges related to task-independent cognitive workload assessment using EEG spatial information. The model employs the differential entropy (DE) and power spectral density (PSD) features of each EEG channel across four frequency bands (delta, theta, alpha, and beta) as node information. For the construction of the network structure, phase-locked values (PLVs), phase-lag indices (PLIs), Pearson correlation coefficients (PCCs), and mutual information (MI) are utilized and evaluated to generate a functional brain network. Specifically, the model aggregates spatial information on the dynamic map by stacking the graph attention layers and utilizes the convolution module to extract the frequency domain information from between the networks under each frequency band. We conducted a cognitive workload experiment with 15 subjects and selected three representative psychological experimental task paradigms (N-back, mental arithmetic, and Sternberg) to induce different levels of cognitive workload (low, medium, and high). Our framework achieved an average accuracy of 65.11% in recognizing the task-independent cognitive workload across the three scenarios. Full article

In this paper, we consider the application of brute force computational techniques (BFCTs) for solving computational problems in mathematical analysis and matrix algebra in a floating-point computing environment. These techniques include, among others, simple matrix computations and the analysis of graphs of functions. Since BFCTs are based on matrix calculations, the program system MATLAB^® is suitable for their computer realization. The computations in this paper are completed in double precision floating-point arithmetic, obeying the 2019 IEEE Standard for binary floating-point calculations. One of the aims of this paper is to analyze cases where popular algorithms and software fail to produce correct answers, failing to alert the user. In real-time control applications, this may have catastrophic consequences with heavy material damage and human casualties. It is known, or suspected, that a number of man-made catastrophes such as the Dharhan accident (1991), Ariane 5 launch failure (1996), Boeing 737 Max tragedies (2018, 2019) and others are due to errors in the computer software and hardware. Another application of BFCTs is finding good initial guesses for known computational algorithms. Sometimes, simple and relatively fast BFCTs are useful tools in solving computational problems correctly and in real time. Among particular problems considered are the genuine addition of machine numbers, numerically stable computations, finding minimums of arrays, the minimization of functions, solving finite equations, integration and differentiation, computing condensed and canonical forms of matrices and clarifying the concepts of the least squares method in the light of the conflict remainders vs. errors. Usually, BFCTs are applied under the user’s supervision, which is not possible in the automatic implementation of computational methods. To implement BFCTs automatically is a challenging problem in the area of artificial intelligence and of mathematical artificial intelligence in particular. BFCTs allow to reveal the underlying arithmetic in the performance of computational algorithms. Last but not least, this paper has tutorial value, as computational algorithms and mathematical software are often taught without considering the properties of computational algorithms and machine arithmetic. Full article

The rapid advancement of knowledge graph (KG) technology has led to the emergence of temporal knowledge graphs (TKGs), which represent dynamic relationships over time. Temporal knowledge graph embedding (TKGE) techniques are commonly employed for link prediction and knowledge graph completion, among other tasks. However, existing TKGE models mainly rely on basic arithmetic operations, such as addition, subtraction, and multiplication, which limits their capacity to capture complex, non-linear relationships between entities. Moreover, many neural network-based TKGE models focus on static entities and relationships, overlooking the temporal dynamics of entity neighborhoods and their potential for encoding relational patterns, which can result in significant semantic loss. To address these limitations, we propose DuaTHP, a novel model that integrates Transformer blocks with Householder projections in the dual quaternion space. DuaTHP utilizes Householder projections to map head-to-tail entity relations, effectively capturing key relational patterns. The model incorporates two Transformer blocks: the entity Transformer, which models entity–relationship interactions, and the context Transformer, which aggregates relational and temporal information. Additionally, we introduce a time-restricted neighbor selector, which focuses on neighbors interacting within a specific time frame to enhance domain-specific analysis. Experimental results demonstrate that DuaTHP significantly outperforms existing methods in link prediction and knowledge graph completion, effectively addressing both semantic loss and time-related issues in TKGs. Full article

Chemical graph theory connects the analysis of molecular structures with mathematical graph theory, allowing for the prediction of chemical and physical properties through the use of topological indices. Among these, the recently introduced Harmonic–Arithmetic (HA) index, proposed by Abeer M. Albalahi et al. in 2023, offers a novel method to quantify molecular and graph structures. It is defined as $HA\left(G\right)={\sum }_{\mu \omega \in E\left(G\right)}{\displaystyle \frac{4d_G\left(\mu \right)d_G\left(\omega \right)}{{(d_G\left(\mu \right)+d_G\left(\omega \right))}^2}}$, where $d_G\left(\mu \right)$ and $d_G\left(\omega \right)$ are degrees of nodes $\mu$ and $\omega$ in G. In this paper, the HA index examines the bounds for a tree T of order n, with a maximum degree $∆$. The application of the HA index extends to QSPR/QSAR analyses, where topological indices play a crucial role in predicting the relationship between molecular structures and physicochemical properties, such as in Parkinson’s, disease-related antibiotics by calculating their topological indices and analyzing them using QSPR models. Comparative analyses were performed between linear regression models and curvilinear-approach quadratic and cubic regression models to identify the minimal RMSE and enhance predictive accuracy for physicochemical properties. The results demonstrate that the HA index effectively connects mathematical graph theory with molecular characterization, offering reliable predictions, dependable bounds for tree graphs, and meaningful insights into molecular properties. These findings highlight the HA index’s potential as a versatile and innovative tool in advancing chemical graph theory and its applications to real-world problems in chemistry. Full article

The subject of this paper is the development of rationalized algorithms of discrete sinusoidal transform of type I for short sequences of length N = 2, 3, 4, 5, 6, 7, and 8. Here, by the word “rationalization”, we mean the reduction of the number of arithmetic operations required to implement the algorithms. The arithmetic complexity of the developed algorithms is presented in the final table. For each algorithm, we also provide data flow graphs demonstrating the space–time structure of the computational processes. The algorithms were tested to verify their validity using MATLAB software (version R2023). Full article

Consider a unicyclic graph G with edge set $E\left(G\right)$. Let $\mathfrak{f}$ be a real-valued symmetric function defined on the Cartesian square of the set of all distinct elements of G’s degree sequence. A graphical edge-weight-function index of G is defined as ${\mathcal{I}}_{\mathfrak{f}}\left(G\right)={\sum }_{xy\in E\left(G\right)}\mathfrak{f}(d_G\left(x\right),d_G\left(y\right))$, where $d_G\left(x\right)$ denotes the degree a vertex x in G. This paper determines optimal bounds for ${\mathcal{I}}_{\mathfrak{f}}\left(G\right)$ in terms of the order of G and a parameter $\mathfrak{z}$, where $\mathfrak{z}$ is either the number of pendent vertices of G or the matching number of G. The paper also fully characterizes all unicyclic graphs that achieve these bounds. The function $\mathfrak{f}$ must satisfy specific requirements, which are met by several popular indices, including the Sombor index (and its reduced version), arithmetic–geometric index, sigma index, and symmetric division degree index. Consequently, the general results obtained provide bounds for several well-known indices. Full article