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%% Comprehensive Test Template for the utmj LaTeX Class
%% Version 1.0 - Use this file to test layout, metadata, and multi-author macros.

\documentclass{utmj}

\nocover % turn this on if we do not want the cover page

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% 1. JOURNAL EDITORIAL METADATA
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\volume{2026}
\issue{X}
\article{- - -}
\pubyear{2026}
\UTMJpagerange{4005--\pageref*{LastPage}}
\doi{https://doi.org/10.7290/utmjXXXXXX}

% Editorial tracking dates
\datereceived{January 15, 2026}
\daterevised{April 20, 2026}
\dateaccepted{May 10, 2026}

%==== Journal Editors
%\editorinchief{ABCDEFG}

%\managingeditors{DD \quad\rule[-0.2ex]{0.5pt}{2ex}\quad  CC} 


%\guesteditors{Hongjie Dong \quad\rule[-0.2ex]{0.5pt}{2ex}\quad Hung Tran \quad\rule[-0.2ex]{0.5pt}{2ex}\quad Yannick Sire } 



%\editorialboard{\noindent Alex Duon\\  Suzanne Lenhart\\
%  Tadele Meghesha\\
%  Shashikant Mulay\\
%  Duc Nguyen\\[0.2cm] \columnbreak
%  \noindent Hung Nguyen\\
%  Yulan Qing\\
%  Jan Rosinski\\
% William Vivu\\
%  Vyron Vellis}

%\advisoryboard{
% \noindent MM \\ ABC \\  DDD\\
% NN \\[0.2cm] \columnbreak
%\noindent FF \\ GG \\ CDF  \\ DD
%} 


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% 2. ARTICLE METADATA & TITLES
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\title{On the Geometric Convergence of Fractional Invariants in Non-Euclidean Topologies}
\shorttitle{Fractional Invariants in Non-Euclidean Topologies}

\keywords{Fractional invariants, geometric convergence, Riemannian manifolds, topological entropy}
\msc{54A05, 58J20, 37C40}
\email{JRandom@utk.edu (J. Random), jsmith@ornl.gov, rjones@gmail.com (R. Jones), and jsmith@vanderbilt.edu (J. Smith)}



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% 3. MULTI-AUTHOR SYSTEM WITH GRANTS & AFFILIATIONS
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% Author 1 (With Short Name & Grant Footnote)
\UTMJauthor[1][J. Random]{Joe  Random\orcid{0000-0002-...}}
\UTMJaddress{1}{Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA}

% Author 2 (With Short Name & Different Grant Footnote)
\UTMJauthor[2][J. Smith]{Jane Smith\orcid{0000-0002-...}}
%\address{2}{Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA}

% Author 3 (Omitted bracket short-name test to verify automatic fallback logic)
\UTMJauthor[2][R. Jones]{Robert Jones\corrauth\orcid{0000-0002-...}}
\UTMJaddress{2}{Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA}

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% 4. MAIN DOCUMENT CONTENT
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\begin{document}

\maketitle

\begin{abstract}
This paper establishes an analytical framework for calculating the geometric convergence rates of fractional invariant measures on complete Riemannian manifolds. By extending classical localization techniques to localized fractional laplacians, we prove that the structural stability of topological entropy remains invariant under bounded metric perturbations. Our findings generalize several foundational theorems in non-Euclidean dynamics and provide new algorithmic lower bounds for multi-scale numerical integration schemes.
\end{abstract}

\section{Introduction}
Let $(\mathcal{M}, g)$ be a complete, smooth Riemannian manifold with non-positive sectional curvature. The study of invariant measures under the action of fractional diffusion operators has seen significant development. In this paper, we address the open problem regarding structural stability limits when the metric $g$ undergoes local variations.

The layout of this template tests your exact geometry choices ($1.2\text{ in}$ margins, $0.75\text{ in}$ vertical buffers). Notice that subsequent page headers automatically display the parsed author string or short title without clipping into the printer's boundary box.

\section{Core Theoretical Results}
We begin by establishing the necessary analytical bound requirements to verify that your class file handles structural mathematical formulas alongside standard mathematical theorem counters cleanly.

\begin{definition}
A fractional invariant measure $\mu$ is said to be \textit{geometrically regular} if there exists a localized tracking constant $\Lambda > 0$ such that for all test functions $\phi \in C_c^\infty(\mathcal{M})$:
\rule[-1ex]{0pt}{2.5ex} % Subtle vertical padding matrix helper
\begin{equation}
    \int_{\mathcal{M}} (-\Delta)^\alpha \phi \, d\mu \le \Lambda \|\phi\|_{L^2(\mathcal{M})}.
\end{equation}
\end{definition}

With our basic definitions established, we present the principal limiting convergence theorem below.

\begin{theorem}[Main Convergence Property]
\label{thm:main}
Let $\mu_n$ be a sequence of geometrically regular fractional invariant measures on $(\mathcal{M}, g)$. If the structural tracking capacity satisfies $\lim_{n\to\infty} \Lambda_n = \Lambda_\infty$, then $\mu_n$ converges weakly to a limiting measure $\mu_\infty$ satisfying the metric condition:
\begin{equation}
|\mu_n(E) - \mu_\infty(E)| = \mathcal{O}\left(n^{-\frac{2\alpha}{d}}\right).
\end{equation}
\end{theorem}

\begin{proof}
The proof proceeds via standard localization techniques. Let $\Omega \subset \mathcal{M}$ be a compact subset. By applying the fractional Sobolev embedding theorem directly to our kernel matrix, we deduce that the local trace operator is completely bounded. The remaining convergence constraints follow immediately from standard density arguments.
\end{proof}

\begin{lemma}
Under the assumptions of Theorem~\ref{thm:main}, any bounded perturbation of the background manifold metric $g$ preserves weak compactness properties.
\end{lemma}

\begin{remark}
Note that if $\alpha \to 1$, the convergence rate smoothly degenerates back into the classical local operator bounds established by classical diffusion models.
\end{remark}

\section{Numerical Verification and Benchmarking}
To verify how multi-column floats or layout matrices interact with your document text block width ($6.1\text{ in}$), numerical test suites were compiled using local cluster instances. The resulting text flows cleanly around the mathematical frameworks.

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% 5. JOURNAL-WIDE ACKNOWLEDGMENTS
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\begin{acknowledgments} J. Random was supported by NSF, grant \#123456789. 
The authors express their gratitude to Dr. X. Fang and the University of Tennessee for the hospitality. Computing resources were graciously provided by the Advanced Scientific Computing Research infrastructure cluster.
\end{acknowledgments}

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% 6. REFINED MANUAL BIBLIOGRAPHY BLOCK
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\begin{thebibliography}{99}

\bibitem{nguyen2026} 
T. Nguyen, \textit{An Advanced Proof in Topology}, 
Univ. Tenn. Math. J. \textbf{5} (2026), no. 2, 12--45.

\bibitem{smith2024} 
J. Smith, \textit{Foundations of Non-Euclidean Geometry}, 
Academic Press, New York, 2024.

\bibitem{jones2025} 
R. Jones and S. Wise, \textit{Fractional Laplacian Perturbations on Riemannian Manifolds}, 
Journal of Functional Analysis, \textbf{288} (2025), no. 4, 102--129.

\bibitem{O} E. \"{O}zsoy, {\it Geophysical Fluid Dynamics I: An Introduction to Atmosphere - Ocean Dynamics: Homogeneous Fluids}, Springer Textbooks in Earth Sciences, Geography and Environment (2020).

\bibitem{Ped} J. Pedlosky, {\it Geophysical Fluid Dynamics}. Society for Industrial and Applied Mathematics (2005).

\bibitem{SC-1} M. Sammartino and R. E.  Caflisch, {\it  Zero viscosity limit for analytic solutions, of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations}. Comm. Math. Phys. 192 (1998), no. 2, 433-461. 

\bibitem{SC-2} M. Sammartino and R. E.  Caflisch, {\it Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution}. Comm. Math. Phys. 192 (1998), no. 2, 463-491.

\bibitem{Swan} H. Swann, {\it The convergence with vanishing viscosity of non-stationary Navier-Stokes flow to ideal flow in $\mathbb{R}^3$}, Trans. Amer. Math. Soc. 157 (1971), 373-397.

\bibitem{Tarfulea}  A. Tarfulea, {\it Improved a priori bounds for thermal fluid equations}. Trans. Amer. Math. Soc. 371 (2019), no. 4, 2719-2737.

\bibitem{Temam-1} R. Temam, {\it On the Euler equations of incompressible perfect fluids}, J. Func. Anal. 20 (1975), 32-43.

\bibitem{Temam-2} R. Temam, {\it Navier-Stokes Equations}, North-Holland, 1983.

\bibitem{Wang} F. Wang, {\it 
The three-dimensional inviscid limit problem with data analytic near the boundary}. 
SIAM J. Math. Anal. 52 (2020), no. 4, 3520-3545.


\end{thebibliography}

\end{document}
