Graph-Based Fixed Points in 𝐹-Metric Spaces: A New Approach 🔢🔗 | #Sciencefather #researchers #Metric

🔗 Fixed Points & Contractions in $\mathrm{<strong\; data-end="259"\; data-start="197"\; style="display:\; inline\; !important;"><span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord\; mathnormal">F</span></span></span></span>-Metric\; Space\; Graphs\; 📈🔢</strong>}$$\mathrm{<strong\; data-end="259"\; data-start="197"\; style="display:\; inline\; !important;\; font-weight:\; bold;"><br></strong>}$

$Fixed point theory is a cornerstone of modern mathematics, influencing fields like functional analysis, topology, differential equations, and numerical analysis. In this study, we explore new fixed point results in F-metric spaces equipped with graph structures, presenting novel contraction conditions that extend classical results to more generalized spaces.$

 Mathematical Foundation: Fixed Points in F-Metric Spaces

In standard metric spaces (X,d), a function f:X→X has a fixed point x∗ if:

f(x∗)=x∗

A fundamental result in this area is Banach’s Contraction Principle, which guarantees the existence of a unique fixed point for contractive mappings. However, this principle does not directly apply to F-metric spaces, where the metric function takes a more generalized form:

F:X×X×X→R+

subject to specific axioms, generalizing distance measurement beyond traditional metrics.

This study extends fixed point theorems to F-metric spaces by introducing graphic contractions, where mappings satisfy contractive conditions along the edges of an associated graph G(X,E). This combination of graph theory and fixed point analysis enables us to study convergence in spaces with complex connectivity patterns.

 Graph-Based Contraction Mappings

Consider a directed graph G(X,E), where:

• Nodes represent elements of the F-metric space X.

• Edges define the contraction conditions between pairs of points.

A mapping f:X→X is G-contractive if there exists α∈(0,1) such that:

$                                                                                                                            F(f(x),f(y),z)≤α⋅F(x,y,z)∀(x,y)∈E$

where F satisfies specific contraction properties. The graph constraint allows us to refine classical results, ensuring convergence even when the entire space is not contractive.

 Real-World Applications: Fixed Points & Graph Theory

Fixed point theory is essential in solving nonlinear equations, optimization problems, and dynamic systems. Some key applications include:

 Mathematical Modeling  – Used in complex systems with non-traditional distances.
 Computational Methods  – Convergence of iterative algorithms in machine learning.
 Physics & Engineering  – Stability analysis in chaotic and dynamical systems.
 Network Theory  – Used in graph-based data structures and communication networks.

 Solving Fractional Differential Equations (FDEs) with Fixed Points

Fractional calculus extends classical derivatives to non-integer orders, leading to fractional differential equations (FDEs) of the form:

Dαy(t)=f(t,y(t))

where Dα represents the fractional derivative. These equations model memory-dependent and non-local processes in physics, biology, and finance.

By applying F-metric space fixed point results, we establish existence and uniqueness criteria for solutions to boundary value problems involving FDEs. This generalization provides new insights into long-term stability and convergence in systems with fractional dynamics.

 Future Directions & Mathematical Impact

This research bridges fixed point theory, graph structures, and fractional calculus, paving the way for new advances in:

 Nonlinear Analysis & Functional Equations 
 Mathematical Optimization & Numerical Methods 
 Graph-Based Computational Models 

By extending fixed point results to F-metric spaces with graphs, this study contributes to the broader mathematical landscape, opening new doors for mathematical modeling, applied analysis, and theoretical advancements.

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🔢 Math-Driven Insights into H₂/CO₂ Adsorption 🌡️📊 | #Sciencefather #researchers #AppliedMathematics

🧮 AI + Mathematics = Next-Gen Gas Adsorption 🚀

🔢✨ This study fuses mathematical modeling and 🤖 AI-driven predictions to unveil H₂/CO₂ adsorption dynamics on CuBTC & MIL-125(Ti)_NH₂. By blending empirical formulas 📈 with deep learning 🧠, it optimizes gas separation, advancing hydrogen purification ⚡ and carbon capture 🌍, revolutionizing adsorption efficiency with cutting-edge computation! 🚀📊

🔢 Breaking the Code of H₂/CO₂ Adsorption!

What happens when mathematics, artificial intelligence, and chemistry collide? Breakthrough curves—the secret signatures of gas separation—hold the answer! These curves, governed by differential equations, adsorption isotherms, and AI predictions, are the keys to unlocking efficient gas adsorption. In this study, we redefine adsorption science by fusing math-driven models with deep learning intelligence to predict H₂/CO₂ behavior in advanced materials like CuBTC and MIL-125(Ti)_NH₂.

📐 The Mathematical DNA of Adsorption

Adsorption isn’t random—it follows a precise mathematical script! The journey of H₂ and CO₂ molecules inside porous materials is dictated by:

➗ Partial Differential Equations (PDEs): Tracking gas diffusion and adsorption kinetics.
📊 Langmuir & Freundlich Isotherms: Decoding how gases bind to solid surfaces.
⚙️ Linear Driving Force (LDF) Model: Measuring the speed of mass transfer.
🔎 Optimization Algorithms: Tweaking adsorption efficiency for maximum performance.

Mathematics turns the chaos of gas separation into predictable, optimizable science!

🤖 AI: The Ultimate Adsorption Mathematician

Traditional equations lay the foundation, but deep neural networks (DNNs) take it further:

🧠 Learning Beyond Equations: AI detects adsorption behaviors that math models miss.
⚡ Supercharged Predictions: Faster, smarter, and more precise breakthrough curve forecasting.
🔬 Multivariable Mastery: AI processes temperature, pressure, and molecular dynamics all at once.

By merging equations with AI, we create a self-learning mathematical framework that evolves with real-world data!

🌍 Why It Matters: Real-World Applications

🔬 Hydrogen Purification: Refining H₂ recovery for a sustainable future.
🌱 Carbon Capture Revolution: Smarter CO₂ adsorption for cleaner air.
🚀 AI-Optimized Gas Separation: Powering industries with AI-enhanced mathematical models.

The synergy of math and AI is revolutionizing gas adsorption—one equation at a time!

🧮 Conclusion: The Equation for the Future

Mathematics is no longer just about solving for x—it’s about solving for the future! With AI-powered differential models, adsorption isotherms, and deep learning, we’re writing a new equation for innovation in gas adsorption science. This isn’t just math—it’s the future of clean energy and smart materials!

🔢 + 🤖 = 🚀 Mathematical Breakthroughs in Adsorption Science!

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"Mathematics-Driven 3D Gaussian Splatting for Smarter Urban Wind Simulations" 📐🌆💨 | #Sciencefather #researchers #AppliedMathematics

🚀 Transforming Urban Wind Simulations with 3D Gaussian Splatting 🌆💨

🌟 The Perfect Blend of Math, AI, and CFD!

Imagine a world where entire cities 🏙️ can be reconstructed from just a few drone images—accurate, fast, and CFD-ready! Thanks to 3D Gaussian Splatting, we can now transform sparse point clouds into high-fidelity urban models optimized for wind flow analysis. This cutting-edge approach merges probability theory, matrix transformations, and fluid dynamics to revolutionize urban planning, energy efficiency, and wind comfort studies. 🌍✨

🧠 The Math Behind 3D Gaussian Splatting 📊

At its core, 3D Gaussian Splatting assigns each point in a cloud a probability distribution, smoothing irregularities and filling in missing data. The Gaussian function, which defines this transformation, is:

$$G(\mathbf{x}) = \frac{1}{(2\pi)^{3/2} |\Sigma|^{1/2}} \exp\left(-\frac{1}{2} (\mathbf{x} - \mathbf{\mu})^T \Sigma^{-1} (\mathbf{x} - \mathbf{\mu}) \right)$$

🔢 Why It’s Game-Changing?

✅ Smooth & Accurate: Converts noisy point clouds into realistic 3D structures 🎭
✅ Speed Boost: Reduces complexity while keeping precision ⚡
✅ Seamless Integration: Works directly with AI and CFD systems 🤖

This smart mathematical filtering ensures that every detail—from skyscrapers to small alleys—is captured with high fidelity! 🏗️✨

📐 Matrix Transformations: Building a Smarter City Model 🏗️

Once we generate high-quality point clouds, we need to align, scale, and refine them for CFD simulations. This is done through rigid and affine transformations:

🔹 Rigid Transformations (Rotation + Translation):

$$\mathbf{x'} = R\mathbf{x} + \mathbf{t}$$

📌 Aligns the model with real-world coordinates! 🎯

🔹 Affine Transformations (Scaling, Shearing, Rotation):

$$\mathbf{x'} = A\mathbf{x} + \mathbf{b}$$

📌 Optimizes building shapes for accurate CFD-ready geometry! 🌍

These transformations ensure that our city models match real-world dimensions with pixel-perfect accuracy. 🔍

💨 Cracking Urban Wind Flow with Navier-Stokes Equations 🌪️

Once we have a detailed 3D city, we need to simulate how air flows through buildings. The Navier-Stokes equations govern this airflow:

$$\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{F}$$

✅ Analyzes turbulence & wind speed 🌬️
✅ Optimizes urban design for better airflow 🏙️
✅ Reduces wind discomfort & improves ventilation 🌿

These equations help predict wind patterns, ensuring safer, more comfortable urban environments. 🚶💨

🚀 Why This Approach is a Game-Changer?

✅ 3-5× Faster than traditional 3D modeling methods 🏎️
✅ 12% More Accurate in point cloud reconstruction 🎯
✅ LoD2 & LoD2.5 Detail Levels for high-precision simulations 🔍
✅ Grid Convergence Index (GCI): 3.76% ensuring CFD stability 📊

With Gaussian Splatting, we’re reshaping the future of urban wind analysis, creating greener, smarter, and wind-optimized cities! 🌍💨

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"📊 Mathematical Model of UV Degradation in Brine 🌊" | #Sciencefather #researchers #AppliedMathematics

Mathematical Modeling of Octadecylamine and 4-Dodecylmorpholine Degradation in Salt Lake Brine Under UV Light

UV light accelerates the degradation of octadecylamine and 4-dodecylmorpholine in salt brine. 📉 The process follows first-order kinetics, modeled by differential equations. 🔢 Quantum yield measures efficiency, while salt ions and pH influence reaction rates. 🌊 Statistical analysis predicts degradation trends, ensuring accurate environmental impact assessment. 🌞✅

1️⃣ Rate of Degradation and Kinetic Equations 📉

• The degradation follows first-order kinetics, meaning the rate depends on the concentration of the compound: $$\frac{dC}{dt} = -kC$$
• Solving this gives the concentration at time $t$: $$C_t = C_0 e^{-kt}$$
• $C_0$ = initial concentration, $C_t$ = concentration at time $t$, and $k$ = degradation rate constant.

2️⃣ Half-Life Calculation ⏳

• The time for the compound’s concentration to reduce by 50% is: $$t_{1/2} = \frac{\ln 2}{k}$$
• A higher $k$ value indicates a faster degradation process.

3️⃣ UV Absorption and Quantum Yield 🌞🔢

• The efficiency of UV radiation in degrading the molecules is given by: $$\Phi = \frac{\text{Molecules degraded}}{\text{Photons absorbed}}$$
• Higher quantum yield ($\Phi$) means faster and more efficient photodegradation.

4️⃣ Influence of Salt and Environmental Factors 🌊📊

• The degradation rate is modified by ionic strength, pH, and UV intensity: $$\frac{dC}{dt} = -kC f(\text{pH, } I_{UV}, \text{ionic strength})$$
• $I_{UV}$ represents UV intensity, affecting how quickly bonds break.
• Regression analysis is used to determine the best kinetic model based on experimental data.

5️⃣ Graphical and Statistical Analysis 📈

• A first-order reaction shows a linear relationship when plotting $\ln C$ vs. $t$.
• A second-order reaction is linear when plotting $1/C$ vs. $t$.
• Correlation coefficients (R²) help identify the most accurate kinetic model.

✅ Conclusion 🔬

• Mathematical models predict how quickly octadecylamine and 4-dodecylmorpholine degrade under UV light.
• Using differential equations, quantum yield, and regression analysis, we can estimate environmental impact and optimize degradation conditions.

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Transpiration Cooling of N₂–He Mixture in Hypersonic Flow 🚀🔥 | #Sciencefather #researchers #AppliedMathematics

Numerical Analysis of Transpiration Cooling Using N₂–He Binary Mixture in Hypersonic Laminar Flow 🚀🔥

Hypersonic vehicles face extreme heat due to atmospheric friction. Transpiration cooling injects gas through a porous surface, forming a protective thermal barrier. While helium is highly effective, its storage volume is impractical. A helium-nitrogen mixture (8% He, 92% N₂) reduces reservoir size by 78.5%, maintaining peak thermal performance. This breakthrough enhances efficiency, compactness, and safety in futuristic aerospace designs.

1. The Hypersonic Heating Challenge

• Hypersonic vehicles travel at speeds beyond Mach 5, facing intense aerodynamic heating.
• Extreme temperatures can melt even the most advanced materials, making cooling systems essential.
• Transpiration cooling injects a gas through porous surfaces, creating a protective thermal barrier.

2. The Problem with Pure Helium Cooling

• Helium is an excellent coolant due to its high thermal conductivity and low viscosity.
• However, its low molecular weight requires large storage volumes, making it impractical for compact aerospace systems.
• Engineers needed a solution to maintain efficiency while reducing storage size.

3. The Optimal Cooling Solution: Helium–Nitrogen Mixture

• A binary gas mixture of helium (8%) and nitrogen (92%) was identified as the best alternative.
• Key findings from numerical simulations:
  ✅ Thermal effectiveness = 1.0 (unity), ensuring excellent heat dissipation.
  ✅ Coolant reservoir volume reduced to 285 cm³ (a 78.5% decrease compared to pure helium).
  ✅ Maintains a balance between efficiency and compact storage.

4. Impact on Hypersonic Vehicle Design

• Reduces storage requirements, making it ideal for space-constrained aerospace applications.
• Enhances heat dissipation, preventing vehicle failure at extreme speeds.
• Optimized gas flow ensures smooth transpiration cooling.

5. The Future of Hypersonic Travel

• This breakthrough in cooling technology paves the way for:
  ✅ Faster, safer, and more efficient hypersonic vehicles.
  ✅ Advancements in aerospace, defense, and space exploration.
  ✅ Enhanced material durability and thermal management systems.

With this innovation, hypersonic travel is becoming more practical than ever. 🚀🔥

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Heartiest congratulations Dr. Peng Yue | #Sciencefather #researchers #BestResearcher

🎉 Heartfelt Congratulations to Prof. Dr. Peng Yue! 🏆🎖️

The Math Scientist Awards proudly recognizes Prof. Dr. Peng Yue as the Best Researcher Award winner! His groundbreaking work in mathematics, mechanics, and computational science has set new benchmarks in the research community.

🔹 Current Position: Professor at Ocean University of China
🔹 Field of Expertise: Mathematics | Mechanics | Computational Science
🔹 Professional Affiliations:

• Member of the Chinese Mathematical Society
• Member of the Chinese Society for Industrial & Applied Mathematics
• Member of the Chinese Society of Mechanics

🎓 Academic Journey & Education:

📍 Ph.D. in Mechanics – National Aerospace University "Kharkov Aviation Institute" (2017)
📍 Master’s in Mechanics – National Aerospace University "Kharkov Aviation Institute" (2013)
📍 Bachelor’s in Theoretical & Applied Mechanics – National Aerospace University "Kharkov Aviation Institute" (2011)

His education laid a strong foundation for his innovative research in fluid dynamics, applied mathematics, and computational mechanics.

🏅 Research & Achievements:

📖 Published extensively in high-impact journals, including:

• Mathematics (2025) – Uncertain Numbers
• Journal of Physics: Conference Series (2023) – Mathematical Model for Excited State Fluid Dynamics
• Symmetry (2023) – Surface Pressure Calculation in Multi-Field Coupling Mechanism
• Physics of Fluids (2021) – Flowfield Separation & Transition Analysis
• AIP Advances (2020) – Boundary Element Method in Aerodynamic Research

🚀 Innovations & Research Contributions:

• Developed advanced mathematical models for nonlinear & complex systems
• Pioneered computational techniques in fluid dynamics & applied mechanics
• Improved surface pressure calculations in multi-field coupling mechanisms

🎤 Keynote Speaker & Conference Participation:

• Keynote Speaker – 26th International Congress of Theoretical & Applied Mechanics (ICTAM 2024)
• Attendee – 13th International Conference on Computational Mechanics (2024)

🏛️ Previous Academic & Research Positions:

• Professor at China Aerodynamics Research & Development Center (2020-2023)
• Researcher & Educator at University of Electronic Science & Technology of China (2019-2023)

🌟 A Visionary Leader in Mathematics & Mechanics!

💡 Prof. Dr. Peng Yue, your passion for advancing scientific knowledge continues to inspire mathematicians, physicists, and engineers worldwide. Your achievements illuminate the path for future researchers, and your dedication pushes the boundaries of innovation! 🚀👏

Wishing you continued success in pioneering groundbreaking research and shaping the future of mathematics and mechanics! 🎊🎉

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"Conjugate Heat Dissipation in Double-Layer Stacked Heat Sink" | #Sciencefather #researchers #mathematics

Double-Layer Mini/Micro-Channel Stacked Heat Sink: A Revolutionary Cooling Solution 🚀🔥

In high-performance electronics, industrial machinery, and computing systems, efficient heat dissipation is a critical challenge. Traditional single-layer microchannel heat sinks often face high thermal resistance and uneven temperature distribution, leading to overheating and performance degradation. To address these issues, an innovative double-layer mini/micro-channel stacked heat sink has been developed. This advanced design enhances cooling efficiency, reduces pressure drop, and ensures better temperature uniformity compared to conventional heat sinks.

Design and Structure ⚙️

The heat sink consists of two distinct layers:
🔹 Top Layer (Mini-Channels): Facilitates high coolant flow, distributing heat evenly.
🔹 Bottom Layer (Micro-Channels): Provides localized cooling with an increased surface area, enhancing heat dissipation.
🔹 Metal Construction (Aluminum/Copper): Ensures superior thermal conductivity and durability.
🔹 Coolant Flow Optimization: Concurrent flow of pure water or phase change nanoemulsion maximizes cooling efficiency.

How It Works 🔄❄️

The cooling process involves:
✅ Heat is generated from the thermal source (e.g., electronic components, processors, or industrial machines).
✅ Coolant enters the mini-channels, absorbs heat, and distributes it uniformly.
✅ Partially heated coolant flows into micro-channels, where further cooling occurs due to increased heat transfer.
✅ The heat-absorbed coolant exits, preventing temperature build-up and enhancing cooling efficiency.

Phase Change Nanoemulsion: The Game Changer 🧪💧

Apart from using pure water, the system employs phase change nanoemulsion (PCN), a coolant infused with nanoparticles that undergo phase transitions (solid to liquid or liquid to gas), absorbing extra heat in the process.

💡 Benefits of Phase Change Nanoemulsion:
🔸 Higher Heat Absorption – Due to latent heat effects.
🔸 More Uniform Temperature Distribution – Reduces hotspots and ensures thermal stability.
🔸 Increased Heat Transfer Coefficient – A 36.14% boost compared to single-layer microchannel heat sinks.

Numerical Simulation and Key Findings 📊🖥️

To validate its performance, the heat sink was tested using:
🔹 Pseudo-vorticity-velocity method for 3D velocity field calculations.
🔹 Finite volume method for solving heat transfer equations.

Key results:
✔️ At a flow rate ratio of 0.5, total flow rate of 25.48 mL/min, and heat flux of 25 W/cm², the overall heat transfer coefficient increased by 36.14% compared to a single-layer heat sink with pure water.
✔️ Higher flow rate ratios with low total flow rates resulted in increased thermal resistance (>1), indicating flow optimization is critical for maximum efficiency.

Applications 🏭🔬

This heat sink is ideal for:
🔸 High-performance computing (HPC) systems 🖥️
🔸 Cooling industrial electronics & power devices ⚡
🔸 Automotive & aerospace thermal management 🚗✈️
🔸 Energy-efficient heat exchangers 🔋

Conclusion 🎯

The double-layer mini/micro-channel stacked heat sink is a breakthrough innovation in cooling technology. With its higher heat dissipation efficiency, lower thermal resistance, and advanced phase change nanoemulsion cooling, this system sets a new benchmark in thermal management.

🌟 Why It Stands Out?
✅ Superior heat transfer efficiency
✅ Enhanced temperature uniformity
✅ Lower pressure drop & thermal resistance
✅ Ideal for next-gen electronics & industrial applications

This cutting-edge design is paving the way for a cooler, more efficient future in heat sink technology! 🚀❄️🔥

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