Papers

Representations of the Kohno-Drinfeld Lie algebra associated with several classical multivariate distributions have played an important role in recent developments in the theory of quantum superintegrable systems. In this work, we analyze the corresponding Gaudin models for the multivariate Hahn, Dirichlet, multinomial, and negative multinomial distributions. More precisely, using tools from representation theory, we construct multivariate orthogonal polynomials with respect to these distributions as common eigenfunctions of Gaudin operators. The polynomials are parametrized by solutions to the Bethe ansatz equations or, equivalently, by the roots of Heine-Stieltjes polynomials.

We investigate groups $\hat{G}$ given by a presentation $\mathcal{P}=\langle \textbf{x}: \textbf{r} \rangle$ whose relators $\textbf{r} \subseteq F(\textbf{x})$ are comprised of a set of subwords in $F(\textbf{x})$, i.e. $\textbf{r}$ admits a $\textbf{u}$-substitution in the sense that there exists a homomorphsim $\epsilon: F(\textbf{u}) \rightarrow F(\textbf{x})$ and a subset $\mathbf{v} \subseteq F(\textbf{u})$ such that $\textbf{r}=\epsilon(\textbf{v})$. Equivalently, $\mathcal{P}= \langle \textbf{x}: \epsilon (\textbf{v}) \rangle$ is referred to as the composition of the presentation $\mathcal{G}= \langle \textbf{u}: \textbf{v} \rangle$ with $\mathcal{H}= \langle \textbf{x}: \epsilon(\textbf{u}) \rangle$ of the groups $G$ and $H$, respectively. We survey known results and record structural properties which do not explicitly appear in the literature, e.g. that there is the relative presentation $\langle G, \textbf{x}: \textbf{u}= \epsilon(\textbf{u}) \rangle$ for $\hat{G}$. Thus there is a natural map $\epsilon: G \rightarrow \hat{G}$ and we may consider the associated problems (e.g. injectivity, finiteness). As an application, we investigate the class of groups $\mathcal{G}(\mathcal{B})$ obtained from substituting a presentation of the trivial group into a deficiency one group presentation for $\mathbb{Z}$. Our results show $\mathcal{G}(\mathcal{B})$ is a proper subset of the class of 2-knot groups and properly contains all classical knot groups. A subfamily is investigated which includes the group of the trefoil knot and uses Higman's presentations for the trivial group.

We prove that every symmetric pair of a connected complex reductive group is regular in the sense of Aizenbud–Gourevitch. This settles the Aizenbud–Gourevitch regularity conjecture over the complex numbers. Generalized Harish–Chandra descent then makes the canonical central cover of every complex symmetric pair a Gelfand–Kazhdan pair. An anti-automorphism arising from a compatible Chevalley involution upgrades the resulting GP2 bound to GP1 on the cover, and finite central descent transfers GP1 to the original pair. In particular, van Dijk's conjecture on complex symmetric pairs follows. Rubio reduced the unresolved irreducible regularity problem to four families: the DIII family $(D_r,A_{r-1}+\mathbb{C})$, the balanced CII family $(C_{2r},C_r+C_r)$, some remaining Spin block pairs, and the EVII pair $(E_7,E_6+\mathbb{C})$. We treat these cases by four different mechanisms. For DIII we construct a sign-equivariant Schwartz distribution on the regular set and extend it across a common orbit boundary by the Chen–Sun theorem. For balanced CII we combine homogeneity, distinguished nilpotent orbits, and a stable-density theorem for the centralizer representation. For Spin blocks we prove pleasantness for unequal odd–odd blocks, use Przebinda's orthogonal-distribution theorem in odd smaller rank, and construct a finite orbit closure with automatic extension in even smaller rank. For EVII we compute the graded-$\mathfrak{sl}_2$ data for all twenty-two nilpotent orbits and use central-torus characters to eliminate the remaining resonances, including the two residual triple-centralizer cases. A finite-component assembly theorem then handles arbitrary connected central quotients and diagonal couplings among simple factors.

Let $\pi(H)$ be the Turán density of an r-uniform hypergraph $H$ and let $H_k^r$ denote the $r$-uniform hypergraph on $r+1$ vertices with exactly $k$ edges, where $1\le k\le r+1$. Sidorenko~(JCT-B, 2024) proved that $\pi(H_3^r)\ge (1.7215-o(1))r^{-2}$ as $r\to\infty$ and $\pi(H_k^r)\ge (C_k+o(1))r^{-(1+1/(k-2))}$ for fixed $k$ as $r\to\infty$. Clemen~later improved the first bound to $\pi(H_3^r)\ge cr^{-2}\sqrt{\log r}$ for some constant $c>0$. In this article, we prove the following results. \begin{itemize} \item For any fixed $\varepsilon>0$, there is a constant $c_\varepsilon>0$ such that $$\pi(H_3^r)\ge \frac{c_\varepsilon}{r(\log r)^{2+\varepsilon}}.$$ %$\pi(H_3^r)\ge 1/(r(\log r)^{2+o(1)})$. Together with the known upper bound $\pi(H_3^r)\le1/r$, this implies $\pi(H_3^r)=r^{-1+o(1)}$. \item For every $3\le k\le r+1$, let $s=\min\{k-2,r-k+2\}$. Then \begin{equation*} 0\le \frac{k-2}{r}-\pi(H_k^r) \le \frac{128}{r}\left(\sqrt{s\log\frac{er}{s}}+\log\frac{er}{s}\right). \end{equation*} This estimate yields several asymptotically sharp results for $\pi(H_k^r)$. For example, $\pi(H_k^r)=(1+o(1))(k-2)/r$ when $\log(er/(k))=o(k)$. \end{itemize}

We investigate the structure of central extensions for algebras in a congruence modular variety. We use a multisorted algebraic object called a clonoid to understand the term clone of such a central extension. We develop the difference clonoid of such a central extension and use it to show that the number of $2$-step nilpotent algebras on a fixed finite set is finite if and only if the set is of squarefree order. The subpower membership problem for a finite algebraic structure $\mathbb{A}$ is the problem of deciding on input $a_1,\dots,a_k, b \in A^n$, whether $b$ is in the subalgebra of $\mathbb{A}^n$ generated by $a_1, \dots, a_k$. We show that for a large class of nilpotent Mal'cev algebras the subpower membership problem is solvable in polynomial time, in particular for $2$-step nilpotent Mal'cev algebras of squarefree order.

We establish time-decay estimates for the two-dimensional fourth-order Schrödinger operator $H=\Delta^2+V$ with a real-valued decaying potential $V$, covering all possible zero-energy threshold obstructions. When zero is a regular point or a first-kind resonance, we prove \[ \left\| H^{\frac{\alpha}{4}}e^{-itH}P_{\mathrm{ac}}(H) \right\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{2+\alpha}{4}}, \qquad -2<\alpha\leq2, \] which matches with the free sharp decay rate throughout the full range of $\alpha$. For a second-kind resonance, the decay rate is $|t|^{-(2+\alpha)/4}(\log(2+|t|))^2$ for every $-2<\alpha\leq2$, with only a logarithmic loss. For the stronger threshold singularities, we show that the large-time behavior is governed by the presence of a \(d\)-wave resonance. If zero is a third-kind resonance, or an eigenvalue accompanied by a \(d\)-wave resonance, we obtain the sharp decay $(\log|t|)^{-1}$ for $\alpha=0$ and $|t|^{-\alpha/4}(\log|t|)^{-2}$ for $0<\alpha\leq2$. If zero is an eigenvalue without a $d$-wave resonance, the second-kind estimate is recovered for $-2<\alpha\leq2$. In addition, in the regular and first-kind resonance cases, we obtainthe logarithmically improved weighted estimate for every $2<\alpha\leq2$ and $s>0$: \[ \left\| \omega^{-s} H^{\frac{\alpha}{4}}e^{-itH}P_{\mathrm{ac}}(H)\omega^{-s} \right\|_{L^1\to L^\infty} \lesssim \frac{1} {|t|^{\frac{2+\alpha}{4}}(\log|t|)^s}, \qquad |t|\geq2, \] where $\omega(x)=\log(2+|x|)$. By contrast, zero is a second-kind resonance for the free operator $\Delta^2$, and the free evolution admits no such logarithmic gain. Thus, in the regular and first-kind cases, the potential changes the zero-energy spectral structure of the free operator, and this change is accompanied by improved weighted decay.

Composite likelihood (CL) methods provide a computationally efficient alternative to full likelihood inference for complex multivariate models by replacing the joint likelihood with a product of lower-dimensional marginal or conditional components. Like the MLE, however, the maximum CL estimator (MCLE) is highly sensitive to data contamination. On the other hand, robust divergence-based procedures such as the minimum density power divergence (DPD) estimator require the full joint density and so scale poorly to complex multivariate models. We introduce the composite DPD (CDPD), a genuine statistical divergence built entirely from the low-dimensional component densities defining a CL, combining the computational scalability of CL with the robustness of the DPD. The resulting minimum CDPD estimator (MCDPDE) robustifies the MCLE without requiring integration over the full multivariate sample space. We establish consistency, asymptotic normality, and the influence function of the MCDPDE under regularity conditions on the component models alone, without requiring correct specification of the full joint distribution. We show that it is qualitatively robust for every positive value of its tuning parameter, unlike the MCLE recovered as the limit. Because its components can be chosen at the pairwise or cell level, the framework guards simultaneously against casewise and cellwise contamination. Operating directly on component densities rather than elliptical distance structures, it extends robust inference beyond the elliptical models to which most existing cellwise-robust procedures are confined. We develop computational algorithms implemented in the accompanying R package mvdpd. Simulation studies and real-data applications show that the MCDPDE achieves substantial robustness gains over the MCLE while retaining competitive efficiency under the assumed model.

We present a potential theoretic approach reducing the analysis of the asymptotic shape of free surfaces to the analysis of a precise ordinary differential equation resulting from the reduction process. Although the approach relies mainly on the principal part of the PDE operator to allow for a representation formula and is thus not restricted to problems of elliptic type, we present it at the clean-cut example of three-dimensional axially symmetric steady incompressible cavity flows, which are Neumann-type Bernoulli free boundary problems and for which frequency formulas are unknown and, if they do exist, insufficient to yield the very precise asymptotic behavior we prove here. In 1946 Norman Levinson derived by a power-law ansatz with a slowly varying correction a precise formula for the asymptotic shape of such cavities. However his result requires very strong assumptions such that it has remained an open problem for 80 years whether the cavity solutions we know to exist by a result by Garabedian-Lewy-Schiffer [12] actually share this asymptotic behavior, or whether at least one solution possessing the Levinson asymptotics exists. Here we answer both questions affirmatively, and we obtain complete rigidity at infinity of the Levinson solution in the class of axially symmetric solutions, that is, any solution satisfying mild and natural assumptions at the fixed boundary and infinity converges asymptotically to the Levinson profile $(\log r)^{-1/4}\sqrt{r}$.

Hard combinatorial optimization problems, many of which are NP-hard, present fundamental algorithmic challenges. Average-case analysis on random instances has emerged as a powerful framework for understanding typical algorithmic performance beyond worst-case guarantees. A substantial body of work has established negative results: for sufficiently hard instances (often controlled by the underlying graph connectivity/constraints density), no known polynomial-time algorithm can significantly outperform naive heuristics in the double asymptotic limit where both problem size and constraints density tend to infinity. We revisit this picture by studying the finite-size behavior of some optimization algorithms across easy, intermediate, and hard regimes. Through rigorous analysis of large-graph asymptotics combined with numerical experiments on canonical problems (maximum independent set and maximum $K$-SAT), we demonstrate that while algorithms do eventually converge to theoretically predicted bounds, this convergence can be remarkably slow. In the intermediate regime where instances are already highly constrained, local algorithms achieve solutions substantially better than their predicted performance in the high-constraint-density limit. This gap between finite-regime and asymptotic behavior has important practical implications: sophisticated algorithmic design remains crucial even when asymptotic theory predicts inevitable failure.

We show that for a discrete isometry subgroup acting on a proper CAT(-1) space X, if the critical exponent is less than $1$, then the critical exponent equals the Hausdorff dimension of the entire limit set. Consequently, the limit set must be a Cantor set. As an application, we prove that any finitely generated, torsion-free discrete subgroup in Isom(X) with critical exponent less than one must be geometrically finite and free. This answers a question of Kapovich.

Additive codes over finite fields generalize linear codes, and additive MDS codes provide a natural extension of linear MDS codes. In this article, we study additive twisted Reed–Solomon (TRS) codes and obtain new constructions of additive MDS codes. First, for additive TRS codes with twist $t=2$ and an arbitrary hook, we establish necessary and sufficient conditions for the codes to be additive MDS, thereby generalizing the results in Section 3 of [Jiayu Ma et al., New families of additive non-Reed-Solomon MDS codes]. In particular, we show that the existence of an additive MDS TRS code with $t=2$ and hook $h=0$ yields codes of larger lengths than those obtained for $t=2$ and $h=k-1$ in [Jiayu Ma et al., New families of additive non-Reed-Solomon MDS codes]. Next, we consider additive TRS codes with twist vector $\mathbf{t}=(1,2)$ and hook vector $\mathbf{h}=(0,0)$, and derive necessary and sufficient conditions for them to be additive MDS. We further establish the existence of such codes. Using the Schur square technique, we obtain mild conditions under which the constructed families are inequivalent to additive Reed–Solomon (RS) codes. Finally, we determine parity-check matrices for both families of additive MDS codes considered in this article.

We study an optimal control problem for a quasilinear parabolic equation modeling nonlinear heat conduction during heat treatment of metals. Under general assumptions, we prove existence of an optimal control and characterize the associated optimal control-state pair in a practically relevant framework. The main difficulty is the quadratic gradient term, which we handle through energy estimates and differentiability properties of the control-to-state map.

We establish the existence of infinitely many Ramanujan-type congruences which hold uniformly for the full $\Z$-module of prime detecting quasimodular forms. In particular, we prove that for every prime $p$, there is at least one arithmetic progression $An+B$ such that every prime-detecting quasimodular form has coefficients divisible by $p$ in this progression.

We consider the following question: if two lattice paths in the same set $\mathcal{D}(a,b)$ have the same Lagrange number, must their band graphs be isomorphic? We exhibit two explicit lattice paths in $\mathcal{D}(17,9)$ with the same Lagrange number but non-isomorphic associated band graphs, thereby giving a counterexample to this question.

This is the first part of a series of papers on the large-degree asymptotics of Gromov–Witten invariants. In this paper, we prove complete large-degree asymptotic expansions, at every fixed genus, for the primary Gromov–Witten invariants of the complex projective plane. The proof uses singularity analysis to transfer the local expansions of generating functions at their dominant singularities to asymptotic expansions of Gromov–Witten invariants. The genus-zero asymptotic expansion is obtained from an analysis of the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation. The higher-genus cases are obtained from the Givental–Teleman reconstruction theorem for semisimple cohomological field theories and from the graph-sum formula for the $R$-matrix action.

We introduce adaptive resolution frames (AR-frames), a class of weighted frames equipped with a partitioned index structure that models multiple resolution levels in Hilbert spaces. We establish a block-triangular representation of the associated operators and prove the convergence of an iterative reconstruction method with explicit error estimates. We further derive optimal block-diagonal preconditioners that minimize the iteration error and obtain stability results showing that AR-frames and their canonical reconstruction operators remain stable under sufficiently small weighted perturbations. These results provide a unified framework for efficient reconstruction and perturbation analysis of AR-frames.

We count Schreier sets that satisfy a neighborhood condition, including $k$-clustered, $k$-consecutive-free, $k$-neighbored, $k$-isolated, and closed under integral $2$-averages. For the first four conditions, we determine the initial counts and prove linear recurrence relations. For the last condition, we prove a recurrence that involves the divisor counting function.

This paper studies integrated sensing and communications (ISAC) over a hybrid system that seamlessly combines legacy cellular base stations with distributed cell-free (CF) access points (APs). We propose a hierarchical ISAC architecture where a central base station (CBS) serves its near users and simultaneously operates as a monostatic radar for aerial target detection, while distributed APs—many idle under user-centric clustering—act as cost-free bistatic receivers. The CBS jointly handles communication processing and multi-static sensing fusion, reducing fronthaul overhead compared to conventional cell-free ISAC. To achieve this, a five-phase time-division duplexing workflow with precise ISAC role assignment is specified. Closed-form expressions for spectral efficiency and multi-static sensing signal-to-noise ratio analytically characterize the communications–sensing Pareto frontier. Numerical results confirm that the proposed hierarchical design simultaneously achieves higher sum throughput and superior sensing accuracy than conventional cell-free ISAC.

This work studies surface-localized transmission eigenfunctions for time-harmonic electromagnetic scattering. Prior results on this phenomenon are mostly qualitative, proving existence without quantifying distribution. Here we provide a quantitative analysis for the Maxwell transmission eigenvalue problem, covering both TE and TM modes. We first establish a Weyl asymptotic law for the full counting function, with cubic growth with respect to the radius $R$. We then prove Weyl-type upper and lower bounds for the counting functions restricted to surface-localized modes, and demonstrate that they share the same growth order three. To our knowledge, this is the first quantitative result of its kind, demonstrating that surface-localized eigenfunctions form a non-negligible fraction of the high-frequency spectrum.

Maximal-density factoring must often make one parent object carry several forms of information at once: child mass, two multiplicity levels, a density estimate, and the ability to survive later geometric refinements. These properties are not stable under arbitrary deletion. We introduce a cellular factoring method that places positive child mass on a weighted bipartite graph of labelled parents and half-open spatial cells. After regularizing edge weights and cell degrees, one edge set defines both the child shading and the parent shading. Compatibility and pointwise multiplicity bounds are then exact, while any later selection of whole parent-cell incidences lifts quantitatively back to the children. The combinatorial core does not require Euclidean geometry: it extends to finite measurable atomic partitions with comparable atom measures and admits an explicit finite-iteration loss. We combine this joint construction with a weighted planar incidence estimate, a resolution-limited cellwise angular selection, a labelled slab estimate, and a global synchronization of tubelet and fine-tube mass. The resulting rectangular parent estimate has shading exponent $2-\sigma$, which is sharp uniformly over the class considered here. With the stated GWZ factoring data and corresponding $K_F(\beta)$ or $K_{KT}(\beta)$, we obtain the two small-middle interfaces used in the reduction. For the very-non-sticky branch, assuming both $K_{KT}(\beta)$ and $K_F(\beta)$, the stated GWZ Lemma 9.1 hypotheses together with the source-uniformity and refinement interface (I1)–(I4) yield the terminal gain for the represented fine-tube input; this conditional application supplies the corresponding input to the surrounding reduction, while the sharp-convex-parent problem lies outside its scope.

Many rigorous results in the modern era of statistical mechanics have been obtained through geometrical representations. A much studied tool in this setting is the random cluster representation of lattice spin models. Another area of statistical mechanics that is of great interest but lacking rigorous results is the theory of the renormalization group. This paper investigates the idea to find a common ground between these two concepts in order to obtain rigorous results on the renormalization group flow. A need for negative cluster-weights weakens the success of this approach. Nevertheless, we managed to establish a relation between the scaling limit of the renormalization group flow of the nearest-neighbour Ising model away from criticality with a simple one-dimensional dynamical system that follows the cluster connectivity of the renormalization group transformation. This allows for a rigorous proof of the convergence of this flow to the zero- and infinite-temperature fixed point respectively for a large family of renormalization group transformations. Explicit results will be established on $\mathbb{Z}^2$ followed by a discussion of the available generalizations to higher dimensions.

Convolution integrals widely exist in applications, and to enable fast and accurate computations, this paper introduces two general multi-stage neural operator learning frameworks. The first, Deep Collocation Neural Operator (DCNO), is a supervised approach that iteratively refines the operator approximation by learning residuals from input-output data pairs. The second, Deep Galerkin Neural Operator (DGNO), is an unsupervised framework applicable when the target operator can be represented by a PDE, leveraging the weak form of the PDE residual for training. Both methods progressively construct basis operators through multiple training stages to enrich the approximation space, leading to significantly improved accuracy over standard one-shot operator learning. We provide theoretical analysis for their approximation capabilities and implement them for learning convolutions. Extensive numerical experiments demonstrate that both DCNO and DGNO achieve high accuracy, approaching machine precision under single float for convolution problems, and offer substantial efficiency gains for numerous queries or parametric variations compared to traditional solvers. We also extend these frameworks to handle multi-input operator learning scenarios involving variations in both the density and kernel of a convolution.

For $g\ge 3$, let $N_g$ denote the non-orientable surface of genus $g$. In this article, we establish the existence of filling pairs on $N_g$ that intersect minimally by construction using the theory of fat graphs. The mapping class group $\mathrm{Mod}(N_g)$ acts on the set of all such filling pairs. We count $\mathrm{Mod}(N_g)$-orbits of this action by providing both lower and upper bounds. Furthermore, we show that both bounds grow super-exponentially with $g$ using graph cohomology. Also, we investigate the lengths of minimally intersecting filling pairs on hyperbolic non-orientable surfaces $X$ in moduli space $\mathcal{M}_g$ of $N_g$. We define a function $\mathcal{F}_g:\mathcal{M}_g\to\mathbb{R}_{>0}$, where for $X\in \mathcal{M}_g$, the function $\mathcal{F}_g(X)$ is the shortest total length of a minimally intersecting filling pair on $X$. We determine its minimum $m_g$ and show that the set of minimizers is in bijection with the \(\mathrm{Mod}(N_g)\)-orbits of minimally intersecting filling pairs. We further extend \(\mathcal{F}_g\) to \(\mathcal{Y}_g\), defined by minimizing the length over all filling pairs, and show that \(\mathcal{Y}_g\) attains the same minimum value as \(\mathcal{F}_g\).

Heretofore, the second and third author conjectured that the D'Arcais polynomials, related to the coefficients of the powers of the Dedekind $\eta$-function, are Hurwitz polynomials except for a root at the origin. We show that this does in fact not hold for a positive proportion of all natural numbers.

We consider a fundamental problem of mixedness testing: Given $n$ copies of an $N$-qubit state $\rho$, determine whether $\rho = \mathbb{I}_d/d$ or $\|\rho-\mathbb{I}_d/d\|_1 \geq \varepsilon$ with high probability, where $d = 2^N$. In particular, we focus on performing this task with simple single-qubit measurements, i.e. measurements that are prepared independently on each qubit. We provide a nearly complete picture of single-qubit mixedness testing by showing that its copy-complexity is $\tilde{\Theta}\left(\sqrt{10}^N/\varepsilon^2\right)$. To establish our lower bound, we introduce a new measurement-dependent lower bound framework for adaptive single-copy state certification. For the upper bound, we present a randomized Pauli basis measurement protocol, which relies on a new primitive for computationally efficient uniformity testing of correlation-concentrated distributions on the Boolean hypercube. In conjunction, we provide lower and upper bounds for mixedness testing with fixed Pauli measurement protocols.

We study a Serrin-type overdetermined problem for the fractional Laplacian in a bounded open set with an isolated non-removable interior singularity. For positive weak solutions of (-Delta)^s u=f(u) in Omega\{0}, u=0 in R^N\Omega, and (partial_eta)^s u=c on partial Omega, with 0<s<1, N>2s and c<0, we isolate a first-order tangential cancellation of the boundary quotient u/delta^s that matches the first-order cancellation needed to close the fractional corner argument and is weaker than requiring u/delta^s in C^1 in a full boundary neighborhood. Under this condition, Omega is a ball centered at the singular point and u is radial and strictly decreasing in the radial variable. No pointwise blow-up rate at the pole is assumed. If Omega is of class C^{2,alpha}, the cancellation is automatic when s>1/2, and for every s in (0,1) when f is constant near zero. In particular, in the torsion case f=1, u(x)=tau_R(x)+kG_R(x,0), where tau_R and G_R are respectively the fractional torsion function and the Green function of the ball.

We study makespan minimisation on two identical parallel machines that share a single server for both loading and unloading. Each job must be loaded, processed without interruption on its assigned machine, and unloaded immediately after processing, with a common positive integer duration for all loading and unloading operations. We prove that the decision problem is NP-complete for every fixed server-operation duration and strongly NP-complete when this duration is part of the input. We then analyse ordinary list scheduling and the longest-processing-time rule in the non-unit setting. List scheduling has a tight supremum ratio of two. For the longest-processing-time rule, we obtain the exact worst-case ratio when all processing times are at least the server-operation duration, and derive new parameter-dependent lower and upper bounds for unrestricted instances. The results show that both processing-time granularity and blocking generated by short jobs shape the approximation behaviour of the common-server problem.

Let $\mathsf{CI}_{\leq c}$ denote the property of being a complete intersection of codimension at most $c$. Although the regular, complete intersection, Gorenstein, and Cohen–Macaulay properties satisfy the Nagata criterion (NC), we prove that $\mathsf{CI}_{\leq c}$ does not satisfy (NC) for any $c \geq 1$. We first construct a counterexample for hypersurfaces and then obtain counterexamples for arbitrary $c$ using square-zero extensions. We also introduce three conditions for a property of Noetherian local rings and show that, under stability with respect to localization and reduction by suitable regular sequences, (NC) is characterized by a lifting property across normally flat nilpotent thickenings. Nevertheless, we recover the expected openness result: the $\mathsf{CI}_{\leq c}$-locus is open for every Noetherian ring satisfying $\mathsf{Reg}$-Q0, and hence for every quasi-excellent ring. Finally, for every quasi-compact excellent scheme $X$, we prove that the subset $\left\{x \in X \mid \operatorname{edim}(\mathcal{O}_{X,x}) - \dim(\mathcal{O}_{X,x}) \leq n \right\}$ is constructible for every $n \in \mathbb{N}$, although the function $x \mapsto \operatorname{edim}(\mathcal{O}_{X,x}) - \dim(\mathcal{O}_{X,x})$ is not upper semicontinuous in general.

Let $D\subset\mathbb R^d$ be a bounded domain, let $\kappa>0$ be fixed, and let $W$ be a fractional Brownian sheet on $\mathbb R\times\mathbb R^d$. Consider the Stratonovich parabolic Anderson model (PAM) $\partial_tu_\kappa=(\frac12\Delta+\kappa W')u_\kappa$ with Dirichlet boundary condition on $D$ and the flat initial condition $u_\kappa(0,\cdot)=\mathbf 1_D$. We calculate exact asymptotics for the expectation and the standard deviation of the total mass $\int_Du_\kappa(t,x)~\mathrm d x$ as $t\to0$ under the assumption that $W$'s Hurst indices are all at least $1/2$ and that $u_\kappa$'s moments are finite for small enough $t>0$. In doing so, we uncover that these asymptotics are determined by a competition between three mechanisms: (1) $\mathbf{Geometry}$: The rate of heat diffusion through the boundary $\partial D$. (2) $\mathbf{Fluctuations}$: $W$'s time Hurst index. (3) $\mathbf{Renormalization}$: The singularity of deterministic Stratonovich corrections. As a result, we identify novel phase transition phenomena, which arise from the influence of $W$'s Hurst indices on the relative magnitudes of these contributions.

For admissible integers $n,\alpha,k$, let $f(n,\alpha,k)$ be the minimum number of edges in a $k$-connected graph of order $n$ and independence number $\alpha$. A conjecture of Bougard and Joret predicts that $f(n,\alpha,k)=\lceil nk/2\rceil$ when $n\leq k\alpha$, under the assumptions $n\geq2\alpha$, $n\geq\alpha+k$, $\alpha\geq2$, and $k\geq3$. We disprove this prediction, determine $f(n,\alpha,k)$ throughout the boundary $n=\alpha+k$, and characterize every extremal graph on that boundary. In particular, for every $k\geq4$, \[ f(2k-1,k-1,k)=k^2-1, \] whereas the conjectured value is $k^2-\lfloor k/2\rfloor$. The extremal graphs in this family are precisely $\overline K_{k-1}\join T$, where $T$ is an arbitrary tree of order $k$. The smallest-order failure has parameters $(n,\alpha,k)=(7,3,4)$, and no admissible counterexample has smaller order.