Papers

We study the branching problem for the pair $SL_{n+m}(\mathbb{C})\supset SL_n(\mathbb{C})\times SL_m(\mathbb{C})$. We describe the corresponding branching rule in terms of a semigroup $\Sigma_{n,m}\subset\Lambda^{+}\times\mathbb{Z}^N$, where $\Lambda^{+}$ is the semigroup of dominant weights of $SL_{n+m}$, and $N$ is the dimension of maximal unipotent subgroup in $SL_{n+m}$. Let $V(\lambda)$ be the irreducible representation of $SL_{n+m}$ with the highest weight $\lambda$. For every dominant weight $\lambda\in\Lambda^{+}$ the set of all $\sigma\in\Sigma_{n,m}$ with dominant weight $\lambda$ parametrizes the irreducible representations in the restriction $V(\lambda)|_{SL_n\times SL_m}$ of $V(\lambda)$ to $SL_{n}\times SL_{m}$. We describe the semigroup $\Sigma_{n,m}$ as the semigroup of integral points in some polyhedral cone and we find the inequalities defining this cone.

Dropout and Random Gradient Masking (RaM) are two training techniques used to improve performance in deep learning. Both techniques inject randomness into the training dynamics, but in significantly different ways: dropout applies random masks to the activations in the forward pass, whereas RaM leaves the forward pass unchanged and instead masks the gradients. In particular, the noise induced by RaM in the parameter updates is unbiased, so standard explanations for the effectiveness of dropout, such as the penalization effect or the prevention of co-adaptation between neurons, do not apply to RaM. In this work, we show that the difference between the two methods disappears for ResNets in the large depth and width asymptotics: in the complete feature learning regime, they both converge to the same large-scale limiting dynamics. This asymptotic equivalence holds for several variants of dropout and RaM, including layerwise dropout as used in stochastic-depth ResNets, albeit at slower quantitative rates. In fact, we also show that several of these variants collapse to the same limit asymptotically.

We show that the inner automorphism groups and automorphism groups of the Cuntz algebras have the same homotopy groups. In particular, the homotopy types of the inner automorphism groups and the projective unitary groups of Cuntz algebras are different.

A general algebraic surface cannot be embedded in P4. Proving a conjecture by Hartshorne and Lichtenbaum, Ellingsrud and Peskine showed that there is a degree bound for smooth rational surfaces in P4, and in fact for surfaces not of general type. We give a survey of the classification status and of classical and computer-aided constructions of smooth non-general type surfaces in P4.

We observe that on a compact almost Hermitian 8 manifold with totally skew-symmetric Nijenhuis tensor parallel with respect to the unique almost hermitian connection with torsion three form if the three Ricci tensors of this connection vanish then the torsion 3-form is closed. Consequently, on a compact complex 8-manifold if the three Ricci tensors of the Strominger-Bismut connection vanish then its torsion is closed, i.e. the space is pluriclosed. We also deduce that a compact ACYT 8-manifold with paralle Nijenhuis tensor with respect to the torsion connection has closed torsion if and only if the trace of the exterior derivative of the torsion vanishes. It is shown that a compact ACYT 8-manifold wich Nijenhuis tensor is parallel with respect to the torsion connection is a Ricci flat $SU(4)$ instanton exactly when the torsion 3-form is parallel with respect to the torsion and to the Levi-Civita connections simultaneously. We consider an almost hyperhermitian 8-manifold with an $Sp(2)$ structure admitting linear connection preserving the $Sp(2)$ structure and having totally skew symmetric torsion and call it AHKT manifold. The exact conditions an almost hyperhermitian 8-manifold to be an AHKT are presented and it is shown that an AHKT 8-manifold is HKT exactly when the three Lee forms coincide

We investigate boundedness and regularity properties of weak solutions to a class of generalized logarithmic double phase equations with variable exponents. The considered operators arise from Musielak-Orlicz type energies of logarithmic double phase type and exhibit nonstandard growth features. Under general structural assumptions, we derive a priori boundedness estimates in the subcritical setting and establish boundedness of weak solutions also in the presence of critical growth terms. In addition, we prove global Hölder continuity up to the boundary by means of the De Giorgi iteration scheme, localization arguments, and the frozen functional technique. The obtained results extend several existing regularity results for double phase and related nonstandard growth problems.

We prove the consistency of Zermelo–Fraenkel set theory with the Axiom of Dependent Choices, no Vitali sets and a large fragment of the Boolean Prime Ideal Theorem.

In this paper, we use the spectral surplus $\lambda(G) - \sqrt{m}$ to measure how far $G$ lies above the Nosal threshold, and prove the following edge-spectral supersaturation results for triangles and books. (a) Every graph $G$ with $m\ge 3$ edges and $\lambda(G) \ge 1 + \sqrt{m-2}$ contains at least $m-2$ triangles, with equality if and only if $G = K_3 \vee \tfrac{m-3}{3} K_1$. This can be viewed as the third-layer supersaturation in the jump phenomenon, after the first layer $t(G) \ge \lfloor \tfrac{1}{2}(\sqrt{m}-1) \rfloor$ proved by Ning and Zhai, and the second layer $t(G) \ge \tfrac{m-1}{2}$ by Zhang and Zhai. (b) Every $m$-edge graph $G$ satisfies $t(G) \ge m\bigl(\lambda - \sqrt{m}\,\bigr)$, with equality if and only if $G$ is complete bipartite. Consequently, $\lambda(G) \ge \sqrt{m} + q$ forces $t(G) > q m$ for every real $q > 0$. This is an edge-spectral counterpart of the Lovász–Simonovits theorem, and it improves the Bollobás–Nikiforov bound $t(G) \ge \tfrac13 \lambda(\lambda^2 - m)$ in the range $\sqrt m \le \lambda(G) \le 1.3\sqrt m $. (c) Every $m$-edge Nosal graph $G$ contains a book of size greater than $\tfrac14 \sqrt{m}$. This improves two recent results on the booksize constant: $\tfrac{1}{24}$ proved by Li, Liu and Zhang, and $\tfrac19$ by Zhai, Li and Lou. This narrows the gap toward the conjectured optimal constant $\tfrac13$. (d) Every $m$-edge Nosal graph $G$ contains at least $\bigl(\tfrac{1}{8} - o(1)\bigr) m$ copies of the kite $C_4^+=B_2$, and the constant $\tfrac18$ is best possible. This determines the sharp asymptotic constant for counting $C_4^+$ and strengthens the $\Omega(m)$ bound of Li, Liu and Zhang.

For a quasinilpotent operator $T$ on a Banach space $X$, R. Douglas and R. Yang associated with each nonzero vector $x$ the local resolvent-growth exponent $k_x$, and introduced the power set $\Lambda(T) = \{k_x : x \neq 0\}$. We prove that $1 \in \Lambda(T)$ for every quasinilpotent operator on an arbitrary Banach space, which answers a question of Ji and Zhang. We further show that $\Lambda(T) = [0,1]$ for every backward unilateral weighted shift on $\ell^p$ whose weight sequence is strictly decreasing and $p'$-summable for some $p' > 0$, thereby weakening the hypotheses imposed by Hu and Ji.

Two graphs are degree-similar if their adjacency matrices and degree matrices are simultaneously similar. Godsil and Sun asked whether non-isomorphic degree-similar unicyclic graphs exist. We prove that they do not exist. Every graph degree-similar to a unicyclic graph is isomorphic to it. The proof uses a symbolic leaf-peeling algorithm to recover the rooted trees attached to the unique cycle, and a rigidity lemma for colored cycles to identify automorphisms of the cycle.

We introduce approximate expressions for the thermodynamics of the two-dimensional Ising model in an external magnetic field. The external field breaks the system's symmetry, which complicates the exact calculation of the free energy. To address this, we write the model in its quaternion representation to apply a small angle approximation with respect to a reference frame. The approximation simplifies the transfer matrix to a centrosymmetric matrix for which the largest eigenvalue is known and depends on the external field strength. We therefore write an approximate expression for the system's free energy and derive the magnetization, susceptibility, internal energy, and specific heat. These results are compared with numerical simulations performed with the BKL algorithm. We find that the approximate expressions and numerical simulations agree in regions where the approximation is valid.

We construct a new type II finite-time blow-up mechanism for the energy-supercritical heat equation \[ u_t=\Delta u+u^3, \qquad n\geq 5. \] The solution is positive and blows up only at the origin, but in a highly anisotropic fashion. As $t\nearrow T$, the solution concentrates in a thin tubular region around an $(n-4)$-dimensional sphere whose radius shrinks to zero at the self-similar scale \[ \xi_r(t)\sim \sqrt{2(n-4)(T-t)}. \] At the same time, concentration takes place transversely to the sphere at the much smaller scale \[ \lambda(t)\sim \kappa_* \frac{T-t}{|\log(T-t)|^{\frac n{n-2}}}, \] for some $\kappa_*>0$. More precisely, in cylindrical coordinates $r=|x'|$, $z\in\mathbb R^3$, the leading profile is \[ u(x,t) \sim \frac{1}{\lambda(t)} U\left( \frac{r-\xi_r(t)}{\lambda(t)}, \frac{z}{\lambda(t)} \right), \] where $U$ is the Aubin–Talenti bubble in $\mathbb R^4$. The construction reveals a two-scale singularity mechanism in which a critical transverse bubble concentrates around a geometric set that itself collapses. The concentration tube evolves at the parabolic scale $\sqrt{T-t}$, whereas its transverse thickness is governed by the much smaller type II scale $\lambda(t)$. The logarithmic blow-up law is determined by a nonlocal modulation equation arising from the interaction between the four-dimensional critical bubble and the axisymmetric heat kernel. To our knowledge, this seems to be the first Type II blowup that quantifies the effect of a self-similar collapsing tube. The exponent $p=3$ is energy-supercritical in dimensions $n\geq5$, but lies below the Joseph–Lundgren exponent for $5\leq n\leq 12$, in a regime where positive radial type II blow-up is ruled out. The present result provides the first example of a positive type II, single-point blow-up through a collapsing thin-tube geometry.

Let $\pi=\{ 2, q \}$ where $q$ is an odd prime. Let $G$ be a finite group of order divisible by a prime $p \in \pi$. We show that the principal $p$-block of $G$ contains a nontrivial irreducible character of degree not divisible by $2$ nor $q$ and with field of values contained in the $q$th cyclotomic extension. This statement simultaneously provides a principal block version of results of Navarro–Tiep and Giannelli–Hung–Schaeffer Fry–Vallejo.

We develop and analyze neural stochastic residual methods for fully nonlinear parabolic terminal problems under exact diffusion sampling. A quadratic stochastic residual supports global training and backward regression with scalar networks and losses involving only spatial derivatives. For global training, the correction removes a Hessian penalty from the limiting first-order loss. Under a uniform output regularity budget, we bound the continuous residual norm by the square roots of the global loss or normalized backward-regression excess loss, an $O(h^{1/2})$ discretization term for time step $h$, and terminal error. Residual stability then gives value, gradient and Hessian error bounds. Under suitable solution and terminal assumptions, we construct admissible networks with $O(h^{1/2})$ bounds on these approximation errors and the square roots of both loss criteria. In ten-dimensional backward-regression tests using a shared schedule tuned for the quadratic residual, mean relative Hessian errors are $5$–$6\%$ with the quadratic residual and about $15\%$ with the first-order residual. Tuned backward regression achieves Hessian accuracy comparable to physics-informed neural networks across the tested nonlinear coefficients. Translation tests show that the ratio of Hessian error to residual can grow sharply under a fixed sampling law.

It is well-known that solutions to the incompressible Navier-Stokes system are limits in various contexts (weak or strong), of solutions to the Boltzmann equation, when the Mach and the Knudsen numbers go to zero. In particular the case of smooth solutions is by now rather well understood. Recent works have aimed at choosing initial data in function spaces as close as possible to those corresponding to well-posedness for the incompressible Navier-Stokes system. This paper tackles the case of measurevalued initial vorticity, in two space dimensions: in the special case when the initial vorticity is a Dirac mass, it is known that the unique solution to the Navier-Stokes system is the solution to the heat equation. We prove that the kinetic emanation of this initial vorticity (slightly smoothed out) leads to a solution of the Boltzmann equation which diverges in a strong way and in a very small time layer, from the expected hydrodynamic limit.

We develop an algebraic transfer calculus for the oriented Schatten profiles of kernels underlying operator-valued Gaussian chaoses. A dimension-free link inequality propagates profile bounds through cut factorizations, tensor products, coefficient maps, and ordered contractions. Besides the constant-one strong Schatten theorem, we prove the sharp weak endpoint \[ \mathfrak{S}_{r,\infty}\times\mathfrak{S}_{r,\infty} \longrightarrow \mathfrak{S}_{r,\infty}(\log\mathfrak{S})^{-1/r}, \qquad 1<r<\infty. \] The estimate holds uniformly over all contracted Hilbert spaces, and the exponent $1/r$ cannot be decreased within the displayed $q=\infty$ Lorentz–Zygmund scale. A separate finite-complexity argument gives sharp effective-rank and finite-cut-rank logarithmic bridges from all-cut operator profiles to Gaussian operator norms. Combined with oriented-flattening Gaussian estimates, the calculus yields continuous multiplication on completed Wick chaoses with noncommuting coefficients, an associative algebra of factorially weighted analytic Wick series, and a local-to-global theorem for loop-free Peter–Weyl fusion trees. We then apply the method to singular Wick multipliers on groups of polynomial growth. For second-order multipliers we obtain sharp necessary and sufficient Schatten convergence thresholds; on $\mathbb{Z}^{D}$ we determine the full singular phase diagram at every order. Fourier transfer gives exact Sobolev, Schatten-class, compactness, and trace-class thresholds for sandwiched Wick multiplication operators on $\mathbb{T}^{D}$, together with sharp Fourier–Galerkin rates and approximation-number decay.

This paper reviews the generalised Fried conjecture for Morse–Smale flows on compact Riemannian manifolds. We first establish the proper framework by constructing the twisted de Rham complex and deriving the Hodge decomposition, which underpins the definition of the Ray–Singer torsion. On the dynamical side, we characterise Morse–Smale vector fields, introducing the Ruelle Zeta function to encode the spectral data of closed orbits and {reviewing the Thom–Smale complex for purely gradient flows. In general, the local contributions of the critical elements of the flow are globally assembled via a Smale filtration to construct the Milnor metric} on the determinant line of the twisted cohomology. Finally, under the assumption of a unitary representation, we present the theorem proving the conjecture: the Ray–Singer metric coincides with the Milnor metric. Furthermore, we discuss the structure of the Milnor metric, which factors into the Ruelle Zeta function evaluated at zero and a specific contribution from the fixed points. We clarify that this object reduces to either the classical algebraic Thom–Smale torsion or the Ruelle Zeta function solely in the limiting cases of purely gradient or fixed-point-free flows.

Non-uniform scaling control enables a multi-agent formation to adjust its shape by compressing or stretching independently along different coordinate axes through inter-agent interactions, offering high flexibility in complex environments. The fundamental idea is encoding the desired formation shape as the kernel of a matrix-valued Laplacian. In open multi-agent systems, however, changes in number of agents, number of edges, and leader selection dynamically alter this Laplacian, destroying the required spectral properties: positive semidefiniteness, correct kernel, and positive definiteness of the follower block (we summarize these properties as the formation spectrum). In this paper, we develop distributed protocols to strategically adjust partial weights of the Laplacian matrix for formation control in arbitrary dimensional space. By implementing the protocols, the desired formation spectrum can be preserved under dynamic topology changes including agent joining, edge addition, agent leaving, and edge removal, while any pair of agents can serve as leaders. Unlike existing Laplacian design methods for affine formation control under topology changes, the proposed approach requires a sparser sensing graph, avoids a predefined parent-child hierarchical structure, and supports leader reassignment. The effectiveness of the proposed protocols is validated through both theoretical analysis and numerical simulations.

We consider a class of infinite critical tree-indexed random walks on $\mathbb Z$, where the motion of particles is subject to vertex reinforcement. We mainly focus on the strong reinforcement regime, where we expect the process to localize almost surely on two sites. Part of our analysis includes the study of a time-dependent generalized Pólya urn process, where the number of draws at each step is prescribed by a sequence $(\sigma_n)_{n\ge 1}$ of arbitrary positive integers, and the probability to pick a ball of a given color is proportional to a function of the number of balls of that color. In particular for bounded sequences $(\sigma_n)_{n\ge 1}$, we recover Rubin's characterization for the fixation of one color.

Let $a(n)$ and $b(n)$ be arithmetic sequences, and $$A(s)=\sum_{n\ge1}a(n)n^{-s}, \qquad B(s)=\sum_{n\ge1}b(n)n^{-s},$$ be the two Dirichlet series related by a certain functional equation. Let $m$ be the analytic degree of the functional equation. For $x>0$ and a positive integer $N$, Friedlander and Iwaniec (2005) define the sharply truncated nonlinear dual sum $$\mathcal B_{\ell,D}(x,N) := \sum_{\substack{n\in\mathbb N\\ n\le N}} b(n)n^{-\beta_m} \cos\left( 2\pi m\left(\frac{nx}{D}\right)^{1/m} +\frac{\pi\ell}{4} \right),$$ where $D\ge1$ is the conductor, $\beta_m:=\frac{m+1}{2m}$, and $\ell=m-3-2k$ is determined by the archimedean weight $k$ of the functional equation. Their Conjecture 1 predicts that, for every $\varepsilon>0$, $$\mathcal B_{\ell,D}(x,N) \ll_{\varepsilon,\boldsymbol\kappa} (DNx)^\varepsilon,$$ uniformly in the variables $x$ and $N$, with the degree, conductor, and archimedean datum fixed. We give counterexamples to this prediction with $$A(s)=B(s)=\zeta(s)^m,\; m\geq 4$$ where $$\zeta(s):=\sum_{n\ge1}n^{-s} \qquad(\operatorname{Re}s>1)$$ is the Riemann zeta function.

In this article, we develop an $r-$uniform analogue of the classical Kövári–Sós–Turán inequality. This yields an alternative proof of Erdős's classical upper bound for complete $r$-partite $r$-uniform hypergraphs. More precisely, we establish that for finite non-empty sets $A_{1},\ldots,A_{r}$ with $|A_{1}|\leq\cdots\leq|A_{r}|$ and sufficiently large positive integer $n$, \[\mathrm{ex}(n,\mathbb{K}^{(r)}[A_{1},\ldots,A_{r}])=O\left(n^{r-\frac{1}{|A_{1}|\ldots|A_{r-1}|}}\right).\] Our proof develops an analytic counting framework based on repeated applications of Hölder's inequality and the enumeration of configurations through multiple sums. The argument combines the principle of inclusion–exclusion with a discrete analogue of Fubini's theorem to obtain recursive estimates for extremal quantities. This provides a unified analytic perspective on the generalized Erdős box problem.

Euler's partition theorem was an important step in the establishment of Integer Partitions as a field in its own right. It has attracted many great mathematicians' attention, including Sylvester, and has foreshadowed a number of further results. Another deep result in Partition Theory is the beautiful Lecture Hall Partition Theorem, which is a finite version of Euler's identity. In this work, we introduce the lecture hall length to partitions into distinct parts, and devise a bijection from partitions into odd parts to partitions into distinct parts, which unifies many results together. Various recent theorems regarding the minima excludant and block index become immediate consequences of this bijection. We also pose an interesting conjecture which may uncover a deep relation between Euler's partition theorem and lecture hall partition theorem.

Detection in massive multiple input multiple output (mMIMO) systems suffers from exponential complexity while using optimal decoders like maximum a posteriori (MAP). We propose a belief propagation-based detector based on directional statistics, applicable to mMIMO systems relying on PSK modulations. Thanks to a continuous relaxation of the PSK modulation to the unit circle, and to the use of von Mises parametric representations of the messages to obtain sparse representations of the (generally infinite dimensional) messages, the proposed method allows for approximate detection with a low complexity which does not depend on the PSK modulation order. Extensions of the algorithm to imperfect channel realizations are also present. We quantify the performance and complexity of the proposed approach and compare it with detection algorithms based on Gaussian approximation.

Let $\ell(\Mat_n(F))$ denote the length of the full matrix algebra for a field $F$, i.e. the largest of the least word length needed to span $\Mat_n(F)$, over all generating sets $S$ of $\Mat_n(F)$. Šitov proved the general estimate $$ \ell(\Mat_n(F)) \leq 2n\log_2 n+4n-4. $$ The purpose of this paper is to obtain a log-log saving, and prove that for every $n>1$, $$ \ell(\Mat_n(F)) \leq 2n\log_2 n-2n\log_2\log_2 n+5n. $$ A theorem of Specht gives a word-criterion for unitary similarity of complex $n\times n$ matrices. The trace argument of Freedman–Gupta–Guralnick, as used by Pappacena, shows that any upper bound on $\ell(\Mat_n(F))$ can be used to bound the terseness $\tau(n)$, i.e. the least upper bound for the length of words needed in Specht's theorem. Thus, for $n> 1$, $$ \tau(n)\leq 4n\log_2 n-4n\log_2\log_2 n+10n+1. $$

This study investigates multiscale stochastic Burgers equations with reflection, wherein the slow component is modeled by a stochastic Burgers equation with reflection and the fast component by a stochastic reaction-diffusion equation with reflection. Using the weak convergence approach, we rigorously establish a large deviation principle for the slow component. Key technical tools include the penalization method, carefully constructed stopping times, and a refined adaptation of Khasminskii's classical time discretization scheme.

How deep does a graph neural network need to be on a sparse graph? We study its purest statistical form: node classification on the sparse contextual stochastic block model (CSBM) with average degree $\Delta=O(1)$, whose local weak limit is a broadcast-labelled Poisson Galton-Watson tree. Prior work derived a message-passing classifier $h_\ell$ that aggregates from each vertex at distance $k\le\ell$ the attenuated evidence $2\operatorname{artanh}(\gamma^k t(X_v))$, with $\gamma$ the edge signal and $t$ a bounded likelihood-ratio transform of the feature. We prove that the value of depth is governed by a single number, the Kesten-Stigum ratio $\kappa=\gamma^2\Delta$. Below the threshold ($\kappa<1$), the error sequence is Cauchy at a geometric rate, $|\mathcal{E}(\ell)-\mathcal{E}(\ell')|\le C\kappa^{(\ell+1)/3}$ for all $\ell'>\ell$, so all layers beyond depth $O(\log(1/\epsilon))$ change the error by less than $\epsilon$; conversely, under mild regularity each sufficiently deep layer still flips the decision with probability at least $c\kappa^{\ell/2}$, the empirically sharp exponent. Above the threshold ($\kappa>1$), depth is geometrically productive: $\mathcal{E}(\ell)$ is driven to a branching-process floor of order at most $1/(\kappa-1)$ at any geometric rate $\kappa^{-s\ell}$, $s<1$ (this bound has content only for $\kappa>17$). No local classifier of any depth beats the universal floor $e^{-\Delta}\Phi(-\zeta)$ set by isolated roots ($\zeta$ the feature signal-to-noise ratio), while the first layer provably helps by an explicit total-variation amount. Simulations with an exact belief-propagation baseline on the same trees show that the pairwise rule's error curve is mildly non-monotone in $\ell$, so an optimal finite depth exists (an exact instance is certified in the appendix), while BP saturates strictly faster, at an effective per-layer ratio below $\kappa$ that we identify.

Numerical simulation of droplet dynamics on complex surfaces with varying wettability is of great significance to both engineering applications and fundamental research. However, existing numerical methods still face challenges in accurately capturing interfacial interactions while preserving physical consistency and computational efficiency. In this work, a physically grounded and efficient smoothed particle hydrodynamics (SPH) model is developed for droplet dynamics simulation. To reduce computational cost, a single-phase droplet modeling strategy is employed. At the interface, long-range interactions are approximated using the SPH kernel function, whereas short-range interactions are represented through pressure. Based on this treatment, an explicit relationship between the intermolecular potential energy and the macroscopic surface tension coefficient is further established, thereby reducing reliance on empirical parameter calibration. The proposed method is first validated through static wetting simulations, where the equilibrium contact angles agree well with the Young–Dupré equation. Further simulations of wetting and droplet impact demonstrate that the method is capable of capturing complex dynamic wetting behaviors.

For $N\geq1$, let $K_N$ be the annular closure of $(\sigma_1\sigma_2^{-1})^N.$ We prove that triply graded annular Khovanov homology over $\mathbb F_2$ detects the underlying unoriented annular link $K_N$. If $3\nmid N$, the only ambiguity is overall orientation reversal. If $3\mid N$, the only ambiguity is independent reversal of components; every such reorientation has the same homology, so this is sharp. The proof combines braid detection from the extremal annular grading with a rigidity theorem: the Jones polynomial and exponent sum determine $(\sigma_1\sigma_2^{-1})^N$ up to conjugacy in $B_3$.

Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension $d$ and let $M$ be a finitely generated $A$-module. Assume $M$ has rank $r > 0$. We show that if $M$ is NOT Cohen-Macaulay then $\mu_d(\mathfrak{m}, M) > r$. If further $A$ is unmixed and $\mu_n(\mathfrak{m}, M) \leq 1$ for some $n \geq d$ then we prove $\text{injdim} \ M < \infty$ and $A$ is Cohen-Macaulay.

This paper studies the asymptotic behavior of positive solutions to the boundary Yamabe equation near an isolated singularity when the metric is not conformally flat. In dimensions 3 and 4, we establishes the sharp upper bound and, in the non-removable case, the matching lower bound, which also gives a necessary and sufficient condition for removability. Moreover, every solution with a non-removable singularity is shown to be asymptotically cylindrically symmetric, without relying on a global classification of Fowler-type singular solutions. These results aslo extend the flat half-space theory of Caffarelli-Jin-Sire-Xiong (2014) to non-flat boundary geometries and provide boundary analogues of the interior theories developed by Marques (2008) and Xiong-Zhang (2022).