Abstract Mathematical oncology integrates mechanistic modeling, experimental biology, clinical medicine, and data science to improve cancer detection, treatment planning, and understanding of tumor dynamics. This work provides a systematic review of the field, tracing its historical development from early population- level growth models and the log-kill hypothesis to contemporary spatial, evolutionary, multiscale, and data-driven approaches, supported by a large-scale bibliometric analysis of the discipline’s emergence and thematic clusters. The review surveys mechanistic descriptions of tumor growth, the tumor microenvironment, clonal evolution, and metastasis, before examining models of treatment response, optimal control of therapy, and evolutionary game theory as frameworks for understanding and counteracting drug resistance. Multiscale modeling approaches for coupling molecular, cellular, and tissue-level processes are discussed alongside data-driven methods, including genomics and bioinformatics, artificial intelligence, and mechanistic learning, which combines knowledge-driven mathematical models with data-driven machine learning. Particular attention is given to digital twins and their prospective clinical validation, and to the practical requirements for translating mathematical models into clinical implementation, including model validation, FAIR data principles, and shared benchmarking infrastructure. The review concludes by outlining future directions for the field, emphasizing that sustained interdisciplinary collaboration across mathematics, biology, data science, and clinical medicine, rather than any single modeling paradigm, will determine the extent to which mathematical oncology translates into improved patient care.
Abstract Mathematical oncology has developed into an increasingly active interdisciplinary field, providing quantitative tools for the study of tumor growth, invasion, metastasis, treatment response, and therapy optimization. Within this field, cellular automata form an important class of models. This overview focuses on cellular automata and their extensions in the context of mathematical oncology. We first introduce the necessary mathematical preliminaries and then present the basic formalism of cellular automata, beginning with deterministic models and elementary cellular automata. We continue with a presentation of biologically motivated extensions, including probabilistic cellular automata, lattice-gas cellular automata, Cellular Potts models, and hybrid cellular automata. Particular attention is given to focusing this survey on the reader with some background in mathematics or computer science, who is interested in first discovering the biological applications of the model. Finally, we review selected works from the recent literature.
Abstract Depression is a common and prognostically important comorbidity of chronic heart failure (CHF). Because datasets from cohort studies in this research area are often small and strongly imbalanced, and a ready-to-apply training and evaluation procedure for binary classification in this setting is still lacking, we provide a rigorous methodological overview and elaborate it into a straightforward protocol: leave-one-out cross-validation (LOOCV) with fixed seeds and default library hyperparameters, and a full panel of confusion matrix metrics, including imbalance-adjusted summaries. As an illustration of the chosen methodology, we reformulated the 9-item Patient Health Questionnaire (PHQ-9) screening as a binary classification task and compared three deliberately chosen standard methods—ridge-regularized logistic regression, decision tree, and RUSBoost—on a public cohort published by Cheng and co-authors in 2023. These methods are evaluated at each learner’s default operating point over the range of PHQ-9 binarization thresholds θ ∈ {1, . . . , 15}. RUSBoost obtained the best results at the standard diagnostic PHQ-9 cut-off θ = 10—sensitivity 0.641, specificity 0.619, accuracy 0.621 and a sensitivity– specificity gap of 0.022. As the only learner with built-in imbalance handling, it also consistently outperforms the two observation-weighted baselines on these metrics across the entire range of thresholds. However, absolute sensitivity and specificity remain below clinically ambitious levels for all three models. We attribute part of this performance ceiling to selection bias introduced during cohort acquisition and eligibility filtering.
Abstract The accurate classification of images into graphics-only, text-only, and mixed-content categories is a critical prerequisite for building efficient, content-aware processing pipelines. This initial triage prevents the unnecessary application of computationally expensive operations, such as Optical Character Recognition (OCR), to irrelevant graphical data. To address this challenge, we introduce the Contextualized Vision Transformer (CVT), a novel architecture designed specifically for this nuanced classification task. The CVT addresses the limitations of standard Vision Transformers (ViTs) through three synergistic components. It employs a Learnable Patch Decomposition (LPD) strategy that efficiently extracts patch embeddings. To model complex spatial arrangements, it introduces an Adaptive Spectral Embedding (ASE) module, which replaces static positional encodings with a dynamic, learnable representation. Crucially, a Contextual Feature Gating (CFG) module enhances feature discriminability by adaptively recalibrating patch-level features, selectively amplifying the salient text or graphic regions. Comprehensive K-fold cross-validation demonstrates the robustness and generalizability of the proposed model. The CVT achieves statistically significant improvements in accuracy, precision, and recall compared to state-of-the-art Vision Transformer baselines. Experimental results highlight the effectiveness of this architecture in providing a fast and accurate solution for the vital task of content triage in large-scale visual processing systems.
Abstract This paper introduces a novel framework for understanding intuitionistic values and fuzzy sets through the automorphism of the unit interval [0, 1]. By generalizing the concepts of classical intuitionistic sets using strong negations generated by these automorphisms, we define the φ -intuitionistic values and explore their arithmetic, establishing a comprehensive set of operations based mainly on additive generators of t-(co)norms and their properties. This approach provides greater flexibility to the modelling with these sets.
Abstract Vision Transformers (ViTs) have emerged as a powerful architecture for various computer vision tasks, including writer identification. However, like their CNN counterparts, they are susceptible to performance degradation in cross-script scenarios because their standard self-attention mechanism learns script-specific visual correlations. To overcome this, we propose a novel architecture, the Langevin Resonance Transformer (LRT). The LRT fundamentally redefines self-attention by replacing the abstract mathematical operation with a physically-grounded dynamic simulation. Each image patch is treated as a particle. The core of the LRT is a novel Langevin Attention Layer, where the interaction between pairs of particles is governed by a learnable potential energy function. The net force on each particle is aggregated from all other particles, and its state is evolved according to the Langevin equation, which models motion under both deterministic and stochastic forces. The LRT treats a writer’s style as a physical system defined by an energy landscape. Because it models biomechanics rather than just visual shapes, the resulting representation works regardless of the script. We tested this on the BRS-ID dataset using a custom augmentation strategy. The results show that LRT achieves higher accuracy than standard Vision Transformers and other top-tier models on the cross-script identification task.
Segmentation of the aorta is crucial for various medical analyses, such as the diagnosis and treatment of cardiovascular diseases. This work presents mathematical models and methods yielding a semi-automatic segmentation of the aorta from non-enhanced CT data. Our framework consists of three steps. First, using the minimal path approach, we extract a path within the aorta from two user-supplied points. Then, using 3D Lagrangian curve evolution, we move the initial path to the approximate centerline of the aorta. The centered path is used in the last step to construct the initial condition for the generalized subjective surface method (GSUBSURF). Applying the GSUBSURF method with this initial condition yields an accurate segmentation of the aorta. The segmentation results and the manual segmentations overlap, with a worst-case mean Hausdorff distance of $2.175 \pm 0.605$ mm for a voxel spacing of $0.977$ mm. Using the aorta centerline and segmentation, we define precise regions of interest along the aorta to assess large-vessel vasculitis from patient FDG-PET/CT image data. The application shows promising results, as we demonstrated widespread inflammation throughout the aorta in a patient before treatment. After treatment, we observed a significant reduction in inflammation while accurately identifying the aorta regions where inflammation persisted. These findings also align with those of experienced medical doctors who have worked on the same cases.
This article considers two secret sharing schemes, namely, conjunctive compartmented and disjunctive compartmented. Suppose a group of companies having many compartments is working on a project. In the conjunctive compartmented scheme, a specified number of users from each compartment of every company has to collaborate to access the secret key. While in a disjunctive compartmented scheme, any company with a predetermined number of users from each compartment can get the secret key. The schemes we have proposed use elliptic curves and bilinear pairings. Our schemes are verifiable and efficient. The proposed schemes are illustrated in full detail by concrete examples using SageMath.
Multiple Right-Hand Sides (MRHS) equations represent a mathematical formalism with applications in algebraic cryptanalysis. Solving MRHS equation systems is in general a difficult problem. In this article, we investigate the efficient use of genetic algorithms for solving random sparse MRHS systems. Our experiments suggest that the steady-state selection method with low elitism and mutation rate gives the best results. If the systems are sparse, the system size does not have a significant impact on the success of the algorithm. On the other hand, the method is very sensitive to the system density, with the success rate rapidly declining with increased system density.
The aim of presented paper is to solve the nonlinear geodetic boundary value problem (BVP) by the finite element method (FEM) involving the mapped infinite elements (MIE). In comparison to our previous works, see [ M acák , M. et al .: On an iterative approach to solving the nonlinear satellite-fixed geodetic boundary-value problem . In: IAG Symp. Vol. 142 (2016), pp. 185–192.] and [M acák , M. et al .: A gravity field modelling in mountainous areas by solving the nonlinear satellite-fixed geodetic boundary value problem with the finite element method , Acta Geodaetica et Geophysica, 58 (2023), 305–320.] dealing with bounded domains, in this paper we propose and study numerical concept on unbounded domains, given as an exterior BVP for the Laplace equation outside the gravitating body, e.g. Earth, with the nonlinear boundary condition (BC) prescribed on the Earth’s surface and considering the solution regularity condition at infinity. This concept can be found in many scientific disciplines being also the most natural from physical geodesy point of view, see, e.g., [B ackus , G. E.: Application of a non-linear boundary-value problem for Laplace ’ s equation to gravity and geomagnetic intensity surveys , Q. J. Mech. Appl. Math. 2 (1968), 195–221.] and of large practical importance when we are not able to prescribe BCs on a bounded domain. The proposed concept is based on the iterative procedure, and as the numerical method we have implemented the FEM with the MIE to take into account the regularity of the disturbing potential at infinity. Since the boundary of the computational domain is the discretized real Earth’s surface considering its topography, as finite and infinite elements we have chosen the triangular prisms. We study and verify this numerical approach by a testing experiment with a homogeneous sphere, by the experiment using EGM2008, and finally, we present one detailed numerical experiment with DTU21GRA data.
The paper proposes the numerical schemes applied in the macrophage segmentation process on a quadtree grid, generated adaptively with respect to intensity of the data. Because the data are sparse and, in general, distinctly rectangular images, to generate the mesh the library ‘p4est’ (parallel forest) has been selected. The library offers tools to connect multiple trees into a ‘forest’ (‘4est’), enabling parallel processing (‘p4est’). The segmentation methods used canbe solvedwith PDEs commonly used in image processing, the linear heat equation and the modified SUBSURF model, for which we proposed explicit and semi-implicit schemes based on the finite volume space discretisation. The choice of numerical algorithms is adapted to the way the grid elements are iterated in the environment of the library. In this paper we focus on a single quadtree. From the numerical point of view, the extension to the forest is straightforward. Description of the library’s principles with respect to the grid generation, its elements iteration, refining, coarsening and balancing the grid with the help of so-called callback functions and parallelism can be found in more detail, e.g., in [7], [8].
This paper deals with rotational quadratic hypersurfaces in an n -dimensional Euclidean space. Namely, we explore some basic properties of the intersection of two rotational quadratic hypersurfaces that have a common focus, but are not necessarily confocal. We prove that any such intersection lies in at most two hyperplanes, and we specify the maximum number of its connected components.
This paper is the first of two complementary papers. In this paper, we present a novel algorithm in the field of cryptography, an area that has garnered significant attention from researchers. Our algorithm is based on matrix calculus over finite rings, enabling efficient encryption and decryption processes without requiring additional computational resources. A key feature of this approach is its formulation of a mathematical problem involving the solution of a system of nonlinear equations, which significantly enhances the system’s security and complexity. Furthermore, we provide a detailed mathematical proof of the algorithm’s correctness.
The main goal of the paper is to characterize the families of [ ϱ 1 , ϱ 2 ]-lower superdense subsets of ℝ, which generalize well-known notions of density topology (in our paper, denoted by L 1 ♦ )and T ∗ topology. Dense sets preserve density, i.e., if E ∈ L 1 ♦ , then for every measurable F ⊂ ℝ, E ∩ F possesses the same lower and upper density as F at every x ∈ E ∩ F . On the other hand, elements of T ∗ preserve positive lower density, i.e., if E ∈ T ∗ , F is measurable, x ∈ E∩F and d ( F, x ) > 0, then d ( E ∩ F, x ) > 0. Taking arbitrary 0 ≤ ϱ 1 ≤ ϱ 2 ≤ 1, ϱ 2 − ϱ 1 < 1, we can define subsets E ⊂ ℝ which preserve [ ϱ 1 , ϱ 2 ]-lower density, i.e., if F is measurable, x ∈ E ∩ F and d ( F, x ) is greater or not less than ϱ 2 ,then d ( E ∩ F, x ) is greater or not less than ϱ 1 . We can define four types of superdense sets, but three of them are equal. Even though the definition and properties of [ ϱ 1 , ϱ 2 ]-lower superdensity and T ∗ topology are similar and all of them consist of very big sets, these families are essentially different. In the paper, we focus on basic properties, characterizations of superdense sets and relationships between [ ϱ 1 , ϱ 2 ]-lower superdense sets for different indices [ ϱ 1 , ϱ 2 ]. We apply the notion of [ ϱ 1 , ϱ 2 ]-lower superdensity to find adders of ϱ -lower continuous functions.
We address discrete versions of Bourgain-Morrey spaces. We prove some basic properties and introduce several equivalent norms on these spaces. Finally, we analyze the action of a dyadic version of the Hardy-Littlewood maximal operatoronsuchspaces.
Let X be a Hausdorff topological space and let A be both an F σ and G δ subset of X .Let also f : A → ℝ be a function for which the inverse image of every open subset U ⊂ ℝ is F σ in A . We will prove that there is a linear extension operator ϕ * such that ϕ * ( f ) has the same property on X . An analogous result is proved for the Baire-one function defined on an analogous subset of ℝ. We will also show that the extension map is (with a supremum norm) an isometry. In the second part of the paper, we deal with classical Borsuk’s non-retract theorem. It says that a unit sphere in ℝ n is not a continuous retract of the unit closed ball. We will show that such a unit sphere is a piecewise continuous retract of the unit closed ball.
The summable ideal for a harmonic series is described by the O’Malley density of subsets of the real line generated by sequences of reals. Since this approach leads to a special kind of summable ideals, we investigate the relationship between them and we find conditions under which summable ideals for specific series coincide with one for the harmonic series.
This article is a follow-up of the work done in the papers [10] and [56], where, along with various results on ℐ-convergence, several properties of ℐ-open sets and ℐ-continuous functions were investigated. In this paper, we investigate some properties of ℐ-boundary points of a set in a topological space. We make some observations on ℐ-convergence in the product space. We consider the ideas of an ℐ-hyperconnected space and an ℐ-submaximal space and derive their basic properties.
Let ℳ be a Hilbert C ∗ -module. A linear mapping ψ : ℳ → ℳ is said to be a Hilbert C ∗ -module Jordan homomorphism on ℳ if it satisfies the equation ψ (〈 a, b 〉 a )= 〈 ψ ( a ), ψ ( b )〉 ψ ( a ) for all a, b ∈ ℳ. In thispaper,we show that if ℳ is prime, then every Hilbert C ∗ -module Jordan homomorphism ψ from ℳ onto ℳ is a Hilbert C ∗ -module homomorphism or a Hilbert C ∗ -module anti-homomorphism on ℳ. We also prove a similar result about generalized Hilbert C ∗ -module Jordan homomorphism.
In Dimension Theory, there are “mapping theorems” that establish relationships between the dimensions of a domain and the range of a continuous mapping. Most of the theorems deal with mappings that satisfy additional conditions, such as being closed mappings. In the “environment” of such studies, the dimensions of continuous mappings have also been studied. In this paper, we introduce and investigate a new notion of dimension for continuous mappings between topological spaces, which is closer to the classical definition of the Lebesgue covering dimension of a space. We discuss various results concerning this dimension. Moreover, we present open questions and proposals of new dimensions of continuous mappings for further studies. They are based on known dimensions of continuous mappings between topological spaces.
