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Open research questions in Advanced Mathematical Theories and Applications

109 gap statements mined from Advanced Mathematical Theories and Applications papers in our 4.5M-paper local library, which holds 243 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

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What the literature leaves open

  • The lack of a geometric origin for Newton's Second Law. The mysterious nature of dark energy.

    The Infinite Fractal Descent (IFD) Framework: A Geometric Unification of Quantum Mechanics and Cosmology · 2026 · DOI

  • The lack of a unified explanation for the π-Revolution across different length scales. The need to connect solid-state physics, planetary dynamics, and fundamental cosmology.

    Scale Resonance: From Graphene Lattice to Saturns Hexagon: An Indicator for the Universal Geometric Order of the π-Revolution · 2026 · DOI

  • The paper highlights the need to restrict the sample to the bitoroidal subset of 1272 inner-torus states obtained by quotient by SU (3).

    Gravitational Balance from Galois Negation: A Structural Theorem on Multiplicative Manifolds with Cross-Domain Verification · 2026 · DOI

  • The paper suggests that future research should focus on developing new mathematical frameworks for understanding complex systems. The research recommends exploring the applications of the framework in fields such as physics and engineering. The paper proposes investigating the concentration of Frobenius signatures on a specific alphabet of integers in other contexts.

    Gravitational Balance from Galois Negation: A Structural Theorem on Multiplicative Manifolds with Cross-Domain Verification · 2026 · DOI

  • The paper identifies a need for a framework for the study of finite return and its applications. The paper identifies a need for further research on the implications of the spiral law.

    The Spiral: Fibonacci, Mandelbrot, and the Law of Finite Return · 2026 · DOI

  • Further research could explore the implications of the compact integer-based construction. The possible discrete pattern in G could be further investigated.

    G_from_two_integers. · 2026 · DOI

  • Future research can apply the coupled Leonardo sequence to real-world problems. The sequence can be used to model complex phenomena. Further analysis of the sequence's properties is needed.

    Some properties of the coupled Leonardo sequence · 2026 · DOI

  • The paper suggests further study of the IFD framework and its implications for cosmology and theoretical physics. The framework provides a new perspective on the mechanics of spatial inflow and topological drag.

    The Mechanics of Simultaneity: Gravity as Spatial Inflow and Time as Localized Kinematic Friction within the Infinite Fractal Descent Framework · 2026 · DOI

  • The paper identifies the need for a complete Lagrangian of TVS23 with zero free parameters.

    The Complete Lagrangian of the Structured Vacuum TVS23: Discrete Mathematics, Fractal Hierarchy, and Zero Free Parameters · 2026 · DOI

  • Lack of a complete and self-consistent framework for fundamental physics.

    2026 Frontier Physics and the PFUSRC Universal Rigid Axiom System: Empirical Consistency and Theoretical Predictions · 2026 · DOI

  • The crossing-volume prescription used in the confinement estimate is not yet derived from bilateral first principles. The note identifies the need for a derivation of the crossing-volume prescription from bilateral first principles.

    Infinity Zero - bc meson · 2026 · DOI

  • The standard model of cosmology faces a catastrophic breakdown at the extreme limits of gravitational collapse. The Black Hole Information Paradox is a profound theoretical crisis that requires a novel solution.

    The Infinite Fractal Descent Cosmology: An Exhaustive Analysis of Micro-Macro Equivalence and the Grand Universe Hierarchy · 2026 · DOI

  • The table demonstrates the asymptotic This mathematical progression provides a second-order insight into the preservation of physical complexity. In the IFD cosmology, because the geometry of space itself shrinks proportionally with the matter, the relative distances between internal particles scale down at the exact same exponential rate as the particles themselves.

    The Infinite Fractal Descent Cosmology: An Exhaustive Analysis of Micro-Macro Equivalence and the Grand Universe Hierarchy · 2026 · DOI

  • Prior work has not derived M_P from axioms and the Hurwitz theorem, without free parameters.

    How Pattern Algebra Derives the Planck Mass M_P Without Free Parameters · 2026 · DOI

  • To explore the implications of the IFD framework for propulsion and superluminal travel.

    The Universal Mass Stratification Gradient: Reevaluating Electromagnetic Charge and the Structural Role of the Neutron in the Infinite Fractal Descent Cosmology · 2026 · DOI

  • The paper notes that gravity is a distinct area of physics that is not yet successfully quantized.

    Gravity, Fine-Structure Constant and Natural Units -- some Thoughts based on Dimensional Analysis -- · 2026

  • The most pressing question regarding our work is to find additional practical problems for which the QHT gives quantum advantage.

    Efficient Quantum Hermite Transform · 2026 · DOI

  • Further development of the IFD framework to incorporate additional fundamental forces. Experimental verification of the framework's predictions. Exploration of potential applications in fields such as quantum computing and materials science.

    The Infinite Fractal Descent (IFD) Framework: A Geometric Unification of Quantum Mechanics and Cosmology · 2026 · DOI

  • Further analysis of the fractal structures in various domains, including cosmology, biology, and quantum mechanics. Development of new models for complex systems that incorporate the principles of the Infinite Fractal Descent framework.

    Empirical Verification of Infinite Fractal Descent: Transitioning from Theoretical Modeling to Observable Phenomena · 2026 · DOI

  • Further development of Pattern Algebra (PA) and its applications to physical theories and models. Investigation of the implications of the derived Planck mass M_P for our understanding of quantum gravity and the hierarchy problem in particle physics.

    How Pattern Algebra Derives the Planck Mass M_P Without Free Parameters · 2026 · DOI

  • Further study of the π-Revolution and its implications. Application of the π-World Formula to other fields of physics. Exploration of the scale resonance phenomenon.

    Scale Resonance: From Graphene Lattice to Saturns Hexagon: An Indicator for the Universal Geometric Order of the π-Revolution · 2026 · DOI

  • Further development of PVM for astrophysical applications. Investigation of the implications of the IFD framework for our understanding of the universe.

    The Prime Core Black Hole and the Epoch of Sub-Quantum Rupture: Topological Stability within the Infinite Fractal Descent Framework · 2026 · DOI

  • Current systems fail to strictly distinguish the Universal Cosmos from the 4-Dimensional Universe. Current systems rely on external parameters and artificial assumptions.

    General Outline of PFUS Primitive Field Unified Steady‑State System: Dynamic Evolution, Structural Fusion & Expansion, Information Coding and Origin of Time · 2026 · DOI

  • Experimental tests of PFUSRC’s theoretical predictions. Further development of the PFUSRC framework to resolve inconsistencies in contemporary physics. Application of PFUSRC to new areas of physics and other fields.

    2026 Frontier Physics and the PFUSRC Universal Rigid Axiom System: Empirical Consistency and Theoretical Predictions · 2026 · DOI

  • Derivation of the crossing-volume prescription from bilateral first principles. Application of the bilateral framework estimate to other similar problems.

    Infinity Zero - bc meson · 2026 · DOI

Most-cited papers in Advanced Mathematical Theories and Applications

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Open questions raised by papers listed on this page include: (1) The lack of a unified explanation for the π-Revolution across different length scales (“Scale Resonance: From Graphene Lattice to Saturns Hexagon: An Indicator for the…”, 2026); (2) The paper suggests that future research should focus on developing new mathematical frameworks for understanding complex systems (“Gravitational Balance from Galois Negation: A Structural Theorem on…”, 2026); (3) Further research could explore the implications of the compact integer-based construction (“G_from_two_integers”, 2026); (4) Future research can apply the coupled Leonardo sequence to real-world problems (“Some properties of the coupled Leonardo sequence”, 2026); (5) Lack of a complete and self-consistent framework for fundamental physics (“2026 Frontier Physics and the PFUSRC Universal Rigid Axiom System: Empirical…”, 2026). 109 gap statements have been mined from Advanced Mathematical Theories and Applications papers in our 4.5M-paper local library, which holds 243 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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