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Marwa Ennaceur

Publications and source records attributed to Marwa Ennaceur.

13 recordsLinked to original sources
arXiv2026

Hodge Laplacians on Weighted Simplicial Complexes: Forms, Closures, and Bounded Realizations

We establish operator-norm bounds for discrete Hodge Laplacians on weighted flag complexes of a fixed dimension $n$, whose $k$-simplices are the $(k+1)$-cliques of a weighted graph; essential self-adjointness on natural cores follows, with no completeness or curvature assumption. Dual up/down degrees give Schur-type bounds in every degree. At top degree a unitary conjugation turns $Δ_{n}$ into an adjacency operator on the line-complex plus a diagonal potential, to which Schur's test applies directly. At $n=1$ this is the Anderson--Morley edge-degree bound $\|Δ_{1}\|\le\max\{°u+°v\}$, obtained here on infinite and on weighted graphs, whence $\|Δ_{1}\|\le2d$ in the $d$-regular case. The sharpness analysis is carried out at $n=1$, for the edge block; in higher degrees we prove boundedness and essential self-adjointness, not sharpness. We give a spectral criterion for the attainment of $2d$, together with sufficient conditions: it is attained on every finite $d$-regular bipartite graph, and on every infinite one that is amenable. Amenability cannot be dropped: the $d$-regular tree, $d\ge3$, has exact norm $d+2\sqrt{d-1}<2d$. The standard periodic lattices being amenable, we compute their exact norms from the Bloch symbols: $2d$ in the bipartite cases, against $9$ for the triangular and $16$ for the face-centered cubic lattice, on which it is strict. Finally, an ordered coloring which any countable complex admits reformulates the skew model unitarily on unoriented simplices, the orientation signs becoming a function of the colors alone.

math.SP↗
arXiv2026

Geometric Criteria for Essential Self-Adjointness of Discrete Hodge Laplacians on Weighted Simplicial Complexes

We develop a geometric framework for the essential self-adjointness (ESA) of discrete Hodge Laplacians on weighted simplicial complexes of arbitrary dimension. Three notions of $χ$-completeness are introduced --- \emph{global}, \emph{local by level}, and \emph{local by region} --- which are formally distinct as definitions, with the pairwise logical separations remaining partly conjectural. The main results are: \emph{(i)} ESA of the Gauss--Bonnet operator $D=d+δ$ and the Hodge Laplacian $L=D^2$ on $\bigoplus C_c^i(\Vc)$ under operative geometric hypotheses (existence of cut-offs with finite support and bounded weighted degree), which are implied in particular by each of the three completeness notions; \emph{(ii)} ESA of the individual Laplacian block $L_\ell$ under the corresponding local-by-level hypotheses, via a quadratic-form argument; \emph{(iii)} ESA via a Kato--Rellich coupling decomposition under local-by-region hypotheses, applied to discrete half-spaces in the lattice $\mathbb{Z}^d$ ($d\geq 2$) where the coupling has operator norm $\sqrt{2}$ (graph case, i.e.\ $n=2$ in our convention; see Remark~\ref{rem:n-convention-intro}), with a sufficient condition extending the argument to $n\geq 3$; \emph{(iv)} a divergence criterion $\sum_k w_k=\infty$ guaranteeing ESA of $L$ even in the absence of $χ$-completeness, with a worked example (a polynomial-branching tree) where ESA holds yet $χ$-completeness fails. The relationship between the three notions is partly clarified: global $χ$-completeness implies local $χ$-completeness at every level (Remark~\ref{rem:hierarchy-direct}); the converse non-implications are formulated as conjectures, motivated by uniform energy lower bounds in concrete examples (Section~\ref{sec:examples}). The framework recovers and extends classical results for graphs and triangulations, including the optimal divergence rate of \cite{BGJ}.

math.SP↗
arXiv2026

Study of Rota-Baxter Operators in Matrix $C^*$-Algebras Motivated by Toeplitz Structures, and Applications to Sliding Mode Control

This paper studies Rota-Baxter operators on the matrix $C^*$-algebra $M_n(\mathbb{C})$, motivated by the discrete Toeplitz algebra (whose role is purely heuristic; see Remark~\ref{rem:toeplitz_scope}). We provide a structural classification of such operators compatible with the $C^*$-norm, analyze their induced Lie brackets, and apply them to deform system matrices in discrete-time delayed systems under sliding mode control. Lyapunov-based Bilinear Matrix Inequality conditions, together with a tractable linear reformulation via $Q=X^{-1}$, guarantee asymptotic stability on the sliding manifold and $\mathcal{L}_2$-gain stability. The effective gain from uncertainty $δ$ to state $x$ is $γ/\sqrtμ$ with $μ=λ_{\min}(-\mathcal{M})>0$ determined \emph{a posteriori}; minimizing $γ$ alone does not minimize this bound, which holds under zero extended initial conditions ($V_0=0$). We work under the standing assumption $m=n$ (square actuation); a supplementary non-degenerate example with $m=1$, $n=2$ illustrates LMI feasibility with $Π\neq0$. All algebraic results are proved directly in $M_n(\mathbb{C})$; no infinite-dimensional reduction is used.

math.RA↗
arXiv2026

Local Lie Theory in Quasi-Banach Lie Algebras: Convergence of the BCH Series and Geometric Implications

We develop a local Lie theory for Lie algebras equipped with a quasi-norm, i.e., complete topological vector spaces satisfying a relaxed triangle inequality $\|x+y\|\le \Ctri(\|x\|+\|y\|)$ with $\Ctri\ge 1$. We prove that the Baker--Campbell--Hausdorff (BCH) series converges in a neighborhood of the origin, provided the quasi-norm admits a continuous Lie bracket with finite continuity constant $\Cbracket$. The proof relies on the Aoki--Rolewicz theorem to construct an equivalent $p$-norm satisfying $p$-subadditivity, enabling rigorous Cauchy-sequence arguments in the complete quasi-metric space $(E, d_p)$. This yields a well-defined local Lie group structure via the exponential map. We analyze the geometric deformation induced by the quasi-norm exponent $p\in(0,1]$, showing that it modifies metric properties while preserving the underlying Lie algebraic structure. Numerical estimates of BCH coefficients up to degree $20$, with coefficients defined precisely via Hall--Lyndon basis projection, demonstrate that classical combinatorial bounds are conservative in the presence of algebraic cancellations, allowing significantly larger practical convergence radii in structured algebras. Applications include weak Schatten ideals $\mathcal{L}_{p,\infty}(H)$ for $0<p<1$ and certain Hardy-space operator algebras. \smallskip\noindent\textbf{Remark on the convergence radius.} The Catalan-majorant method yields convergence for $\|x\|+\|y\| < 1/(4\Cbracket)$; the additional factor $\Ctri$ appearing in the combined constant $\Ctotal = \Ctri\Cbracket$ is an artefact of switching to the $p$-norm to establish Cauchyness of partial sums. When the quasi-norm itself is directly a $p$-norm ($\Ctri=1$), no such penalty arises and the radius reduces to $1/(4\Cbracket)$.

math.FA↗
arXiv2025

Spectral Theory of Almost Periodic Banach--Malcev Algebras and Applications to Moufang Dynamics

We introduce almost periodic Banach--Malcev algebras as a non-associative extension of Bohr's classical theory. Our framework is based on the relative compactness of adjoint orbits $\{e^{t\,\mathrm{ad}(x)}(y)\}$, which yields the spectral characterization $σ(\mathrm{ad}(x)) \subseteq i\mathbb{R}$, uniform boundedness of orbit closures in the strong operator topology, and a continuous functional calculus for almost periodic derivations. Compact Malcev algebras -- most notably the imaginary octonions $\mathrm{Im}(\mathbb{O})$ -- provide canonical finite-dimensional examples, and their associated Moufang loops carry strictly periodic flows. We also analyze structural actions on eigenspaces of the Malcev Laplacian as a concrete case study, where the bounded defect operator $S(x,y) \in \mathcal{B}(M)$ quantifies the non-associative correction. While speculative links to non-associative gauge theory are noted, they lie beyond the established mathematical scope. The recent convergence control of the BCH series for special Banach--Malcev algebras \cite{Athmouni2025} provides analytic justification for the local Moufang structure used throughout.

math.DG↗
arXiv2025

Homogeneous Rota--Baxter Operators of Weight~0 on $B(q)$

We give a complete and rigorous classification of homogeneous weight $0$ Rota--Baxter operators on the Block-type Witt algebra $B(q)$, assuming the operator has integral degree $(k,k') \in \mathbb{Z}^2$. A key correction is established in the non+resonant regime $q \ne k'$ with $k \ne 0$: the profile function $g(i) = f(-k,i)$ must satisfy the nonlinear functional equation \[ (i - j)g(i)g(j) = g(i+j+k')\big[(i + k' + q)g(i) - (j + k' + q)g(j)\big], \] which admits only constant, Kronecker-delta, or finite-support solutions. This excludes previously and erroneously claimed families such as non-constant polynomials, exponentials, or nontrivial periodic functions. In contrast, the resonant case $q = k'$ exhibits full flexibility: any profile $g$ is admissible, provided the operator is supported on the single line $m = -k$. The classification is cohomologically exhaustive for generic $q$ (i.e., when $H^1(B(q),B(q)) = 0$), and is applied to derive all homogeneous post-Lie structures and associated Lie algebra deformations.

math.RA↗
arXiv2025

Classification of Homogeneous Odd Rota--Baxter Operators on a Modified Witt-Type Lie Superalgebra

We classify all homogeneous odd (i.e., parity-reversing) Rota--Baxter operators of weight zero on the modified Witt-type Lie superalgebra $W = \langle L_m, G_n \rangle_{m,n\in\Z}$. Our classification shows that nontrivial such operators are highly constrained: either $g \equiv 0$ and $f$ is arbitrary, or $g \not\equiv 0$ forces $f \equiv 0$, and $g$ must take one of several rigid forms dictated by the integer shift $k$ (necessarily odd when $g(0) \neq 0$). We prove that every Rota--Baxter operator on $W$ decomposes uniquely into even and odd homogeneous components; we restrict our attention to the odd case, which yields the full nontrivial structure. Furthermore, we show that all derivations of $W$ are inner, that no Rota--Baxter operator on $W$ is invertible, and we describe the induced super pre-Lie algebra structure together with its cohomological interpretation.

math.RA↗
arXiv2025

Sharp Ascent--Descent Spectral Stability under Strong Resolvent Convergence

We establish sharp stability results for of non--selfadjoint the ascent and descent spectra under strong resolvent convergence (SRS), a natural framework for finite element approximations of non-selfadjoint and singularly perturbed operators. The key quantitative hypothesis is the reduced minimum modulus $γ(T-λ)>0$, which guarantees closed range and enables the transfer of the Kaashoek -- Taylor criteria via gap convergence of operator graphs. At the essential level, B--Fredholm theory extends stability to powers $(T-λ)^m$ provided $γ((T-λ)^j)>0$ for all $1\le j\le m$. We introduce a computable finite-element diagnostic $γ_h = σ_{\min}(M^{-1/2}(A_h-λM)M^{-1/2})$, which serves as a practical surrogate for $γ(T-λ)$ and remains uniformly positive even in convection-dominated regimes when stabilized schemes (e.g., SUPG) are employed. Numerical experiments confirm that $\liminf_{h\to0}γ_h>0$ is both necessary and sufficient for spectral stability, while a Volterra-type counterexample demonstrates the indispensability of the closed-range condition for powers. The analysis clarifies why norm resolvent convergence fails for rough or singular limits, and how SRS-combined with quantitative control of $γ_h$--rescues ascent--descent stability in realistic computational settings.

math.NA↗
arXiv2025

Spectral Theory and Almost Periodic Structures in Hom--Lie Banach Algebras

We develop a systematic functional-analytic framework for Hom--Lie Banach algebras, introducing bounded $α$-twisted derivations and almost periodic elements. Under natural continuity and compactness assumptions, we establish a complete Bohr--Fourier spectral decomposition of such derivations. We prove that the associated almost periodic and ergodic subspaces are not merely topological complements but closed, $α$-invariant subalgebras, stable under the twisted Lie bracket a key structural novelty that enables coherent restriction of the dynamics. We provide explicit constructions of Hom--Banach--Malcev algebras and demonstrate our theory with concrete operator-algebraic applications, including a novel twisted Weyl algebra example, analyzed via the metaplectic representation, where a non-commuting twist enriches the Bohr spectrum from a cyclic group to a two-dimensional lattice.

math.FA↗
arXiv2025

Spectral analysis of The magnetic Laplacian acting on discrete funnels

We study perturbations of the discrete magnetic Laplacian associated to discrete analogs of funnels. We perturb the metric in a long-range way. We establish a propagation estimate and a Limiting Absorption Principle away from possible embedded eigenvalues. The approach is based on a positive commutator technique.

math-ph↗
arXiv2025

Limiting absorption principle for long-range perturbation in a graphene setting

In this paper, we examine the discrete Laplacian acting on a hexagonal lattice by introducing long-range modifications in both the metric and the potential. Our objective is to establish a Limiting Absorption Principle, excluding possible embedded eigenvalues. To this end, we employ the positive commutator technique as our method.

math-ph↗
arXiv2020

Spectral analysis of the Laplacian acting on discrete cusps and funnels

We study perturbations of the discrete Laplacian associated to discrete analogs of cusps and funnels. We perturb the metric and the potential in a long-range way. We establish a propagation estimate and a Limiting Absorption Principle away from the possible embedded eigenvalues. The approach is based on a positive commutator technique.

math.SP↗