Papers

We present an iterative algorithm to compute an arc-length parameterized spline interpolating a set of points. This differs from other methods where the computed spline either does not interpolate the original points or the parameterization is not the arc-length of the returned curves. Our method is applicable in any dimension $D \ge 2$, and we illustrate it with numerical results for plane curves.

In this note, we are interested in the linear Rayleigh-Taylor instability problem for the incompressible fluid with an upper free surface. Using the spectral theory of self-adjoint and compact operator, we rigorously prove the existence of infinitely many normal mode solutions to the linearized equations. Some numerical computations are presented to support our theoretical study.

For every countable compact space $K$, we determine the exact Borel complexity of the isometry class of the Banach space $\mathcal{C}(K)$. As a byproduct, we also determine the precise Borel complexity of the homeomorphism class of a fixed countable compact space $K$, improving earlier results of Cenzer and Mauldin. The above results provide concrete and natural examples of sets with arbitrarily high, still exactly determined, Borel complexity. Moreover, we find a new characterization of those real $L_1$-preduals that are isometric to $\mathcal{C}(K)$ for some zero-dimensional compact space $K$ and we determine the precise Borel complexity of $\mathcal{C}(2^{\mathbb{N}})$.

Polyatomic kinetic models are essential for accurately capturing the thermodynamic behavior of real gases, as internal energy modes significantly influence transport coefficients, relaxation processes, and non-equilibrium effects that cannot be represented by monoatomic models. The polyatomic ESBGK model describes molecular collisions as a relaxation towards a generalized Gaussian distribution with an anisotropic covariance matrix and an exponentially decaying internal energy distribution. We present a new semi-Lagrangian scheme for the polyatomic Ellipsoidal Statistical BGK (ESBGK) model of the Boltzmann equation. The semi-Lagrangian framework, being deterministic and grid-based, removes the time-step restriction associated with the linear transport term by following the method of characteristics. The potentially stiff relaxation term is treated using an implicit A-stable linear multistep method which, owing to the structure of the BGK operator, can be reformulated into a cheap time-stepping scheme. This yields a highly efficient and numerically stable method. The numerical method is asymptotic preserving and stiffly accurate, meaning the scheme asymptotically converges to a scheme for the Euler equations in the vanishing Knudsen limit. In addition, we prove that the first-order scheme, asymptotically converges to the compressible Navier-Stokes equation with correct transport coefficients. Finally, we propose inflow and outflow boundary conditions suitable for BGK-type kinetic equations. We perform simulations of the Fourier and Couette test case to compare the BGK model with Direct Simulation Monte Carlo (DSMC). To conclude, we demonstrate the method on a challenging orifice flow test case with moving boundaries.

For a graph $G=(V,E),$ a set $S \subseteq V$ is called an isolating set of $G$ if the set $V-N[S]$ is independent. The minimum cardinality of an isolating set in $G$ is the isolation number of $G$, denoted by $\iota(G).$ Here we introduce the isolation subdivision number of a graph $G$, denoted by ${\rm sd}_\iota(G)$, as the minimum number of edges of $G$ that must be subdivided, where each edge can be subdivided at most once, in order to obtain a graph with isolation number greater than $\iota(G).$ We show that the new parameter is well defined for any non-trivial graph different from a star and that it can be arbitrarily large. We present the values of this parameter for some elementary classes of graphs and establish some basic properties. We show also that $1\leq {\rm sd}_\iota(T)\leq 4$ for any tree $T$ different from a star and characterize all trees $T$ with ${\rm sd}_\iota(T)=1.$

We study a curriculum-based university course timetabling problem in which the preferences of two key stakeholder groups - lecturers and students - must be balanced while maintaining continuity across semesters in a weekly repeating timetable. While existing approaches typically rely on single-objective formulations or aggregate multiple objectives into a weighted sum, this can obscure the underlying trade-offs between conflicting stakeholder preferences. We therefore propose a multi-objective mixed-integer programming approach that explicitly separates lecturer and student objectives and incorporates timetable continuity by limiting the number of changes, called perturbations, in the time period assignments of selected courses relative to the corresponding semester of the previous academic year. To explore the resulting trade-offs, we develop a multi-objective solution approach based on the lexicographic $\varepsilon$-constraint method, enabling the computation of a representative set of solutions whose images, i.e., their vectors of objective values, cover different regions of the objective space. The approach is evaluated on real-world instances from the Straubing Campus of the Technical University of Munich. The computational results reveal a clear and consistent trade-off between lecturers' and students' objectives across all instances. Moreover, the number of allowed perturbations is identified as a key decision parameter: relaxing this constraint significantly improves timetable quality for both stakeholder groups, although diminishing returns are observed beyond certain thresholds. Overall, the proposed approach provides decision support by generating a diverse set of optimized timetables and enabling a transparent analysis of stakeholder trade-offs and continuity for the practical timetable planning process.

We prove a function field analogue of Duke's equidistribution theorem for CM points, in the setting of Drinfeld–Stuhler modular curves. Our results thus extend, to the Drinfeld setting, both Duke's theorem on the modular curve and S.-W. Zhang's equidistribution in the case of Shimura curves. Equidistribution is reduced via a Weyl criterion to the decay of toric periods, which Waldspurger's formula expresses through central values of automorphic $L$-functions, bounded in Lindelöf-strength form by the Riemann Hypothesis over function fields. We work at arbitrary level structures and in every positive characteristic.

Automated floor plan generation is an important problem in computational architectural design. The goal is to construct a floor plan from user-defined room numbers and door requirements. The user specifies which rooms must share a door and which rooms must not be adjacent. However, these requirements do not determine the exact placement or shape of the rooms. The task is therefore to arrange the rooms in a single floor plan so that all required door connections are satisfied and no rooms overlap. To address this problem, we propose DPLAN (Door Connectivity to Floor Plan Generation), a graph-based prototype that generates floor plans from door and non-adjacency constraints. The framework operates in three stages. First, the user-defined graph is examined and, if disconnected, additional edges are added to connect its components. Second, a bi-connected plane triangulation is constructed to ensure the existence of a floor plan without overlapping rooms or empty spaces. Third, the triangulated graph is transformed into floor plans. For rectangular floor plans (RFPs), separating triangles are removed by modifying edges without adding new vertices, thereby avoiding the creation of extra rooms. For orthogonal floor plans (OFPs), separating triangles are removed by introducing additional vertices, allowing rectilinear room shapes. By enforcing both door and non-adjacency requirements, the framework generates floor plans that satisfy the given constraints. The method is implemented in Python and includes a prototype for interactive constraint specification and floor plan visualization. Currently, the framework supports rectangular plot boundaries. Future work includes support for non-rectangular plots, dimension-based scaling, and circulation modeling.

Decentralized bilevel optimization (DBO) provides a powerful framework for multi-agent systems to solve local bilevel tasks in a decentralized fashion without the need for a central server. However, most existing DBO methods rely on lower-level strong convexity (LLSC) to guarantee unique solutions and a well-defined hypergradient for stationarity measure, hindering their applicability in many practical scenarios not satisfying LLSC. To overcome this limitation, we introduce a new single-loop DBO algorithm called diminishing quadratically-regularized bilevel decentralized optimization (DUET), which eliminates the need for LLSC by introducing a diminishing quadratic regularization to the lower-level (LL) objective. We show that DUET achieves an iteration complexity of $O(1/T^{1-5p-\frac{11}{4}\tau})$ for approximate KKT-stationary point convergence under relaxed assumptions, where $p$ and $\tau $ are control parameters for LL learning rate and averaging, respectively. In addition, our DUET algorithm incorporates gradient tracking to address data heterogeneity, a key challenge in DBO settings. To the best of our knowledge, this is the first work to tackle DBO without LLSC under decentralized settings with data heterogeneity. Numerical experiments validate the theoretical findings and demonstrate the practical effectiveness of our proposed algorithms.

In this paper, we explain how classical $W$-algebras can be obtained as limits of $q$-deformed $W$-algebras in type $A$. To this end, we give a detailed examination of the construction of deformed $W$-algebras via Hamiltonian reduction of deformed affine Poisson algebras. We first derive an affine Poisson vertex algebra (PVA) from the Poisson structure of the deformed affine Poisson algebra. This construction identifies a specific set of generators of the deformed $W$-algebra with generators of the classical $W$-algebra.

We consider ordinary differential equations (ODE) of the form $u''u - (u')^2 = e^{-x}P(u) - 1$, where $P$ is a polynomial. For $P = u^k$, $k = 3,4,6$ this ODE is equivalent to certain degenerate Painlevé III equations. We study whether families of solutions of these ODEs have asymptotic expansions of the form $u(x) = \sum_{k=0}^{\infty} p_k(x+c)e^{-kx}$ for $Re\,x \to +\infty$, where $c \in \mathbb C$ is an arbitrary constant parameterizing the solution family, $p_k$ are polynomials, with $p_0(x) = x$. We find necessary conditions on $P$ for such expansions to exist. Numerical experiments suggest that these conditions are also sufficient, and the expansions are not only formal, but actually provide a series representation of the solutions. Numerical evidence also suggests a conjecture on the nonnegativity of coefficients of the $p_k$.

The s-version of the finite element method (SFEM) enables locally high-resolution analysis by superimposing independently defined finite element meshes. However, domain-decomposed parallelization is nontrivial because complex interactions arise among degrees of freedom distributed over multiple meshes. In this study, we propose a method for constructing a graph structure that uniformly represents interactions among computational points, including both intra- and inter-mesh interactions, based on the overlap of basis-function supports. We apply the proposed graph representation to the B-spline based SFEM (BSFEM), a high-accuracy SFEM formulation previously proposed by the authors. The resulting graph partition enables the consistent assignment of degrees of freedom and elements to processes and the construction of the MPI communication structure, thereby realizing domain-decomposition-based distributed-memory parallelization of BSFEM. To the best of the authors' knowledge, this BSFEM implementation constitutes the first domain-decomposition-based distributed-memory parallelization of an SFEM-based method. Furthermore, as an example demonstrating the utility of the proposed graph representation, we apply cost-weighted graph partitioning in which the matrix-generation costs specific to BSFEM are incorporated into node weights, and demonstrate effective static load balancing that accounts for the nonuniform matrix-generation workload.

Let $A_0,A_1,\ldots,A_n\in M_N(\mathbb{C})$ be Hermitian matrices and let $G_1,\ldots,G_n$ be independent $M\times M$ GUE matrices normalized so that $\|M^{-1/2}G_i\|\to 2$ almost surely as $M\to\infty$. We study the spectral edges and operator norm of $H_M = A_0\otimes I_M + \frac{1}{\sqrt{M}}\sum_{i=1}^n A_i\otimes G_i$. Lehner's formula identifies the right and left edges of the corresponding free semicircular operator as $\rho_+ = \inf_{Z\succ 0}\lambda_{\max}(A_0+Z+\sum_{i=1}^n A_iZ^{-1}A_i)$ and $\rho_- = \sup_{Z\prec 0}\lambda_{\min}(A_0+Z+\sum_{i=1}^n A_iZ^{-1}A_i)$. Assuming $A_i\succeq 0$ for $i\ge 1$ and $M\ge N$, we prove via concentration and minimax duality the finite-dimensional bounds $\mathbb{E}\lambda_{\max}(H_M)\le \rho_+ + 9\sqrt{nN/M}\,\|\sum_{i=1}^n A_i^2\|_{\mathrm{op}}^{1/2}$ and $\mathbb{E}\lambda_{\min}(H_M)\ge \rho_- - 9\sqrt{nN/M}\,\|\sum_{i=1}^n A_i^2\|_{\mathrm{op}}^{1/2}$. With $\rho_* = \max\{\rho_+,-\rho_-\}$, this yields $\mathbb{E}\|H_M\|_{\mathrm{op}}\le \rho_* + 9\sqrt{nN/M}\,\|\sum_{i=1}^n A_i^2\|_{\mathrm{op}}^{1/2}$. For uniformly bounded positive coefficients, bounded $n$, and $N=o(M)$, one obtains $\limsup_{M\to\infty}\mathbb{E}\|H_M\|_{\mathrm{op}}\le\rho$ whenever $\rho_{*,M}\to\rho$. The proof is a matrix-coefficient extension of classical Sudakov–Fernique comparison, combined with a Davidson–Szarek-type singular-value estimate and dual variational formulas for Lehner's edge quantities over density matrices. We also explain why this approach does not extend sharply to signed Hermitian coefficients.

Dense Gaussian networks are degree-four algebraic networks with compact diameter and coordinate-based routing. Their diameter-level broadcast trees are efficient but fragile under node, link, and runtime-discovered faults. This paper develops a runtime recovery framework for dense Gaussian broadcast networks under static node/link faults and mixed faults, plus single-link faults discovered live. The method re-roots the source so known node faults become boundary leaves whenever possible, then filters failed links and repairs gaps by connecting healthy components of the pruned tree. For a selected root with connected healthy component graph, we prove exactly $c-1$ external repair edges are necessary and sufficient. We also prove deterministic single-link repair, give a constant-size boundary-intersection primitive for source selection, derive a link-avoidance exclusion test, and add a local-obstruction bound explaining why high-order cuts vanish as $k$ grows. Experiments over $k\in\{10,25,50,100,200\}$, up to $80{,}401$ nodes, $280{,}000$ static trials, and $15{,}000$ transient trials show 100% recovery for deterministic and bounded regimes, $99.998\%$ for multi-link faults, and $99.963\%$ for heuristic regimes; non-recovered trials are explained by disconnected components or relocation failure. Re-rooting reduces average repair edges by 80–100% versus fixed-source repair. Patched Gaussian-link Noxim replays confirm packet-complete execution and show re-rooting reduces repair edges, components, and depth. A completion-cycle audit separates repair benefit from latency: ablations confirm completion time depends on relocation, scheduling, delivery tail, and selector objective, so the paper claims edge-minimum repair rather than universal completion-cycle dominance.

One-to-all broadcasting in dense Eisenstein–Jacobi (EJ) networks relies on diameter-level spanning trees that fragment when nodes or links fail. This paper introduces the selected triple $(r,\theta,\Kcomp_{r,\theta})$–a chosen root, a chosen EJ coordinate-reduction orientation, and the healthy component graph induced by that choice–as the fundamental unit of analysis for joint node/link fault recovery. The central result is a necessary and sufficient condition: hybrid repair succeeds if and only if the healthy EJ graph $G'=\Ht-\Fv-\Fe$ is connected. When $G'$ is connected, a spanning tree of $\Kcomp_{r,\theta}$ maps to exactly $c-1$ component-crossing repair edges, which is minimum for the selected pruned tree. Deterministic guarantees include: one/two faulty nodes are always placed on the distance-$t$ boundary by re-rooting; a single failed link is either avoided or repaired by exactly one crossing edge; and the repaired depth satisfies $D_{r,\theta}\le 2t+1$ under shallowest-layer entry selection. A 260,000-trial validation campaign confirms 100% recovery and substantial repair-edge reduction over fixed-source repair across five network scales up to $N=120601$ nodes, while global-BFS, near-miss, and cap-sensitivity audits clarify the tradeoff between reachability, forwarding-state changes, and ranked root selection.

Dense Gaussian and Eisenstein–Jacobi (EJ) networks are algebraic interconnection networks with compact coordinate balls, fixed degree, and simple modular addressing. A source-centered coordinate-reduction tree gives a non-redundant one-to-all broadcast in the fault-free network, but processor faults can split the tree into multiple healthy components. Unlike search-based repair methods that require a linear scan of the network to select the repair plan, the certificate selectors introduced here operate in $O(1)$ time and $O(1)$ memory, consulting only the fault coordinates. This paper develops this stronger formulation for the one- and two-fault regime: a constant-time certificate selector. Given only the faulty coordinates, the selector classifies the relative fault geometry, chooses a coordinate-reduction orientation, and returns a bounded ordered set of component-crossing repair edges. For dense Gaussian networks $G_k$, every source-free fault set with $|F|\le2$ is repaired with depth at most $k+2$ and with exactly $c-1$ external component-crossing edges for the selected fault-pruned orientation. For dense EJ networks $H_t$, every one-fault placement is repaired within depth $t+1$, and every two-fault placement is repaired within depth $t+2$, again with exactly $c-1$ external repair edges. Exhaustive strict validation confirms the Gaussian selector over $146{,}156$ one- and two-fault cases for $k=5,\ldots,12$ and the EJ selector over $52{,}395$ cases for $t=2,\ldots,8$, with zero failures in connectivity, acyclicity, exact repair count, or depth bound.

By resolving the singularities of tailor-made orbifolds via twisted families of blow-ups, we construct manifold bundles $M \rightarrow E \rightarrow S^2$. Using tools from real homotopy theory, we show that these bundles determine a free subgroup in $\pi_2(B\mathrm{hAut}(M)_0)$. The proof relies on a generalisation of Sullivan's result, which describes the real homotopy groups of the monoid of homotopy automorphisms $\mathrm{hAut}(X)$ in terms of derivations of the minimal model of $X$, to the monoid $\mathrm{hAut}_A(X)$ of relative homotopy automorphisms. As an application, we prove that the moduli space of torsion-free $\mathrm{G}_2$-structures arising from many generalised Kummer constructions contains a free subgroup of positive rank in its second homotopy group.

We study negative eigenvalues of $n\times n$ stochastic matrices whose off-diagonal support is constrained by a sparse graph. The main tool is a matching-based inertia principle: if $G$ is bipartite with matching number $\mu(G)$, $S$ is a real symmetric matrix supported on $G$ with nonnegative diagonal entries and whose negative index (i.e. number of negative eigenvalues counted with their multiplicities) is denoted by $\nu_{-}(S) $, then \[ \nu_{-}(S)\leq \mu(G). \] In particular, every $n\times n$ nonnegative tridiagonal stochastic matrix $P$ satisfies $ \nu_{-}(P)\leq \left\lfloor \frac{n}{2}\right\rfloor. $ Consequently, after ordering the eigenvalues of $P$ in the decreasing order, we have $ \lambda_{\lceil n/2\rceil}(P)\geq0, \ \text{and hence} \ \lambda_2(P)\geq0, \mbox{ for } n\geq3. $ This gives an all-dimensional strengthening of the previously known $4\times4$ tridiagonal stochastic result. Next, we show that this tridiagonal bound is sharp in every dimension in both reducible and irreducible cases. Finally, we explore some possible extension and raise some open questions.

Minimum action methods provide a powerful framework for analyzing rare transitions in small-noise-driven dynamical systems, but their practical performance is often limited by time truncation and parameter sensitivity in infinite-horizon problems. In this paper, we develop an efficient Laguerre spectral minimum action method (LMAM) for computing quasi-potentials associated with fixed points of dynamical systems. Based on the large deviation framework, the method computes minimum action paths by formulating the problem on a semi-infinite time interval and discretize the temporal direction using Laguerre functions. An appropriate time rescaling strategy is proposed to enhance accuracy and convergence of the Laguerre spectral approximation. To efficiently handle nonlinear terms, we employ an improved procedure for evaluating Laguerre–Gauss–Radau quadrature, which enables stable and accurate double-precision computations with a large number of Laguerre modes. Precise numerical analysis for the linear problem and a local result for the nonlinear case are developed. Numerical experiments including both ordinary and partial differential equations (Allen-Cahn and Navier-Stokes) are presented to illustrate the accuracy and efficiency of the proposed method.

We prove a generalization of Werner's asymptotic formula for the volume of the illumination body of a convex body, which holds on Riemannian manifolds with Ricci curvature bounded from below. The $\delta$-illumination body of a subset of a Riemannian manifold is defined to be the set of all points such that the union of all minimizing geodesic segments joining the point to the set has volume at most $\delta$.

In this paper, we introduce a modified elephant random walk that exhibits a transition from a uniform memory mechanism to a selective amnesic memory mechanism. Using a vector martingale approach, we study the asymptotic behaviour of the walk across different parameter regimes. In the diffusive and critical regimes, we establish almost sure convergence results, laws of iterated logarithm, asymptotic normality of the walk, and the growth rate of mean square displacement. In the superdiffusive regime, we prove an almost sure convergence result and obtain the corresponding mean square displacement rate for the walk. Also, we study some almost sure convergence results for its center of mass. Later, we extend the model by incorporating random step sizes and obtain asymptotic results for it.

The word problem for an Adian inverse semigroup given by the presentation $Inv\langle a,b|a=ba^nb\rangle$, where $n\geq 1$, is decidable.

Let $ (W,S)$ be a Coxeter system, and let $ w\in W$. Let $ [1,w] := \{ x\in W \mid x \leq_{R} w \} $ where $ \leq_{R}$ denotes the weak right order of $ (W,S)$. The element $ w$ is said to have the ancestor property if there is a unique non-trivial involution of maximal length in the set $ [1,w]$. The ancestor property was first defined by Hart and Rowley in \cite{hart2025noteinvolutionprefixescoxeter} where they conjectured that all non-identity elements in a finite Coxeter system have the ancestor property. In an arbitrary Coxeter system $(W,S)$, we show that the ancestor property holds for any non-identity fully commutative element (see \cite{stembridge1996fully} for the definition of a fully commutative element). In particular, since any element of a right-angled Coxeter system is fully commutative, we show that the ancestor property holds for all non-identity elements of a right-angled Coxeter system. Lastly, we also provide an axiomatization of right-angled Coxeter systems as reflection systems with a reflection cocycle that obeys a certain property called the meet intersection condition.

Sums of independent exponential random variables lead to the Erlang distribution, providing a direct probabilistic route from exponential waiting times to the integer-shape gamma law. This paper investigates how this classical construction changes when the exponential waiting-time density is replaced by the $q$-exponential density of nonextensive statistics. Our main result is an analytical asymptotic expression for the outflow of a mass-conserving cascade of reservoirs driven by a $q$-exponential waiting-time kernel. In the critical case $q=5/3$, the large-cascade flow rate converges to a stable Lévy density whose time argument is shifted by a Galilean-type transformation. This shifted Lévy law gives the asymptotic hydrograph of the cascade. We also found that for the entire regime $1<q<2$ the macroscopic dynamics are governed by $\alpha$-stable Lévy laws. This proves that anomalous non-Gaussian dispersion can emerge from pure mass-conserving convolutional chains without invoking fractional derivatives.

We introduce BASIL, a user-friendly desktop application for process optimization. BASIL employs a Bayesian approach, incorporating special acquisition functions that can be used to solve both single and multi-objective optimization problems. It provides a graphical interface that enables users to input their experimental parameters, optimization objectives, and legacy data. This is then used to build surrogate models, which are coupled with acquisition functions to guide and optimize a process towards a desired objective. To facilitate model building, BASIL provides a variety of predefined surrogate model templates. BASIL can be used to optimize any arbitrary experiment or process with known, user-defined input variables, optimization objectives, and defined output.

It is well known that there is a bijective correspondence between the set of positive integer solutions to the Markov equation and the set of Fano triangles mutation equivalent to the Fano triangle of $\mathbb{P}^2$. In this paper, we establish a higher dimensional generalization of this correspondence for arbitrary Fano simplices of any dimension. On the polyhedral side, we introduce a distinguished class of facets, called admissible facets, and show that their number is preserved under facet mutation. As a consequence, facet mutation classes of Fano simplices carry natural exchange graph structures whose valency is equal to the number of admissible facets. On the arithmetic side, we associate to each Fano simplex a weighted Markov-type equation together with a distinguished positive integer solution, and show that the corresponding arithmetic mutations, given by Vieta involutions, are compatible with facet mutations. More precisely, the assignment from Fano simplices to Diophantine data intertwines combinatorial mutations with arithmetic mutations, thereby relating the mutation dynamics of Fano simplices to the arithmetic dynamics of positive integer solutions. Finally, we introduce a piecewise linear transformation on dual polytopes, called a sliding operator, which realizes combinatorial mutation in the dual picture. As applications, we obtain a volume formula for dual simplices in terms of the associated Diophantine data and recover the multiplicity change formula under mutation.

This new book is intended as a first elementary introduction to Topological Data Analysis for mathematics students seeking a rigorous account of the foundations of persistent homology, as well as for computer scientists interested in its theoretical underpinnings. The exposition is as self-contained as possible: all the required background is recalled when needed, and only a few standard results are cited without proof. One section of the book, devoted to monodromy in biparameter persistence (Section 4.4), requires more advanced knowledge of algebraic topology. Persistent homology can be introduced from different perspectives, reflecting the variety of mathematical languages that have shaped its development over the years. Some approaches emphasize the algebraic foundations of the theory, while others highlight its topological essence. In this book, we adopt the latter viewpoint - the one that historically marked the birth of the subject - because we believe it offers both conceptual clarity and pedagogical effectiveness, making it particularly suitable for undergraduate and early graduate students. This book differs from existing introductory texts in several respects. First, it adopts a functional viewpoint: rather than representing data as finite (pseudo-)metric spaces, it treats them as functions encoding the information to be analyzed. This interpretative framework allows data to be viewed as measurable objects and highlights the role of observers and their equivariances in the analysis process. Second, this perspective provides a natural bridge between Topological Data Analysis and machine learning through the theory of Group Equivariant Non-Expansive Operators (GENEOs), which offers a mathematically grounded framework for incorporating symmetries and invariances into learning systems.

This work solves the enumeration problem for weighted bi-colored plane trees with prescribed numbers of black and white vertices, together with prescribed total edge weights at each vertex. An exact closed formula for a particular case is obtained, and a unified algorithmic method for the general case is provided. We then apply this result to two geometric problems. Firstly, we compute the strong Hurwitz number for a special class of branch datum between Riemann spheres with three branched points. This is done by counting the dessins d'enfants that faithfully record such branched covers, and a clear correspondence between dessins and weighted trees in such case. Secondly, we study the geometric moduli space for a special class of extremal Kähler metrics on Riemann sphere (HCMU spheres), with a single conical singularity. We classify and enumerate the connected components of the moduli space with respect to the Gromov-Hausdorff topology. This is based on an efficient representation of these metric surfaces, and a study of Gromov-Hausdorff limits of such surfaces.

We prove an analogue of the classical Donaldson-Uhlenbeck-Yau theorem by using the prescribed Hermitian-Yang-Mills flow. Let $E$ be a holomorphic vector bundle over a compact Kähler manifold $(M,\omega_g)$. Suppose that for every proper coherent subsheaf $F\subset E$, the following inequality holds: $$ deg_{\omega_g}(F)<deg_{\omega_g}(E). $$ Then, for any initial Hermitian metric $h_0$ on $E$ and any positive-definite Hermitian tensor $P\in \Gamma(M,E^*\otimes \overline E^*)$, the prescribed Hermitian-Yang-Mills flow $$ \ \frac{\partial h}{\partial t} = -\Lambda_{\omega_g}\left(\sqrt{-1}\, R^h\right) + P, $$ admits a global smooth solution on $[0,\infty)$. Moreover, as $t\rightarrow\infty$, the flow converges smoothly to a Hermitian metric $h_\infty$ on $E$ satisfying $$ \Lambda_{\omega_g}\left(\sqrt{-1}\, R^{h_\infty}\right) = P. $$ As an application, we establish that on a Fano manifold $M$, for any Hermitian metric form $\omega$ and any positive-definite Hermitian tensor $P\in\Gamma(M,T^{*1,0}M\otimes T^{*0,1}M)$, there exists a unique Hermitian metric tensor $h$ on $T^{1,0}M$ such that $$ \Lambda_\omega\left(\sqrt R^h\right)=P.$$ This may be viewed as an analogue of the Calabi-Yau theorem for Fano manifolds.

The aim of this paper is to account for viscosity, surface tension, and interactions between micro-bubbles in approximating their resonant behavior in the ultrasonic regime. Original asymptotic formulas for the resonance frequencies are derived in terms of the difference in the acoustic impedance at the interface between the gas and the fluid. Both low-frequency resonances (Minnaert resonances) and higher-frequency resonances, i.e., beyond the subwavelength regime, are considered. We also provide a resonant characterization for a system of several micro-bubbles.