Papers

Let \(\mu\) be a self-similar Cantor measure on \(\mathbb R\) associated with a probability weight vector \(\mathbf p\), let \(K=\operatorname{supp}\mu\), and let \(F\) denote the distribution function of \(\mu\). We characterize the points \(x\in K\) at which the local scaling exponent \[ \lim_{\substack{y\to x, y\in K}} \frac{\log |F(y)-F(x)|}{\log |y-x|} \] exists and assumes a prescribed value. The characterization is formulated in terms of the convergence of the ratio between the accumulated logarithmic mass and geometric scales, together with the sublinear growth of the endpoint runs. Unlike the classical ternary case, our approach applies to arbitrary contraction ratios and probability weights. As an application, we construct a subset \(M\subset K\) of full Hausdorff measure on which the local scaling exponent of \(F\) is ${h(\mathbf q,\mathbf p)}/{\chi(\mathbf q)}$ and establish the exact dimension-distortion formula \[ \dim_{\mathrm H} F(A) = \frac{\chi(\mathbf q)}{h(\mathbf q,\mathbf p)} \dim_{\mathrm H} A \] for every \(A\subset M\), where \(\mathbf q\) is the natural probability vector, \(\chi(\mathbf q)\) is the corresponding Lyapunov exponent, and \(h(\mathbf q,\mathbf p)\) is the cross-entropy of \(\mathbf q\) relative to \(\mathbf p\). A three-branch example illustrates the results.

In this paper we focus on the stability of positive semidefinite matrix sensing maps $\Phi_{\mathcal{A}}=(\langle A_i,X\rangle)_{i=1}^m$ where $A_i\succeq 0$, $X\succeq0$ and $\operatorname{rank}(X)\le r$. We introduce the bi-Lipschitz constants of $\Phi_{\mathcal{A}}$ and define the global condition numbers as the ratio of upper and lower Lipschitz constants. We give deterministic universal lower bounds for these condition numbers that depend only on the rank $r$ and on the underlying field. We then show that rank-one Gaussian measurements asymptotically attain these bounds, except for the operator-norm condition number over $\mathbb R$ when $r=1$. For this exceptional case, we establish a sharper universal lower bound and prove its asymptotic sharpness. As an application, we derive stability guarantees for an $\ell_1$-residual PhaseLift-type estimator at the optimal sampling scale. These results identify the intrinsic limits of conditioning in positive semidefinite matrix sensing and connect these limits to stable low-rank recovery.

Given an integer $k\geq2$ and a graph $F$, the $k$-uniform suspension $\mathcal{S}^kF$ is obtained by adjoining a fixed set of $k-2$ new vertices to every edge of $F$. In this paper, we study two extremal problems for suspensions of even cycles. Write $K^k_t$ for the $k$-uniform clique of order $t$. Let $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ and $\mathrm{ex}(n,K^k_t,\mathcal{S}^kC_{2\ell})$ denote the maximum numbers of edges and copies of $K^k_t$, respectively, in an $\mathcal{S}^kC_{2\ell}$-free $k$-uniform hypergraph on $n$ vertices. We prove that, for every $k\geq2$ and infinitely many $n$, \[\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)=\frac{n^{k-1/2}}{(k+1)!}+O(n^{k-1}).\] This extends a folklore result for $k=2$ and, as an immediate consequence, yields the asymptotics of $\mathrm{ex}(n,\mathcal{S}^kC_4)$ for infinitely many $n$, previously established by Mubayi (for all $n$). Furthermore, for every $k\geq2$ and $\ell\in\{3,5\}$, we determine the order of magnitude \[\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})=\Theta(n^{k-1+1/\ell}).\] This generalizes both the classical graph case $k=2$ and a previous result of Mukherjee for $k=\ell=3$. The principal difficulty in both problems lies in constructing the lower bounds. Our construction for $\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)$ incorporates a novel block-packing structure, which yields substantially more copies of $K^k_{k+1}$ than previously known constructions. For $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ with $\ell\in\{3,5\}$, we establish a natural $k$-uniform version of Wenger graphs, addressing the subtleties involved in lifting extremal graph constructions to suspensions. We also give applications of our results to Turán problems for simplicial complexes.

We prove a strict space separation between static and ordinary dynamic approximate membership at every fixed error rate. For each fixed $\varepsilon\in(0,1)$, a capacity-$n$ ordinary dynamic filter over a universe of size $u$, with zero false negatives, pointwise false-positive probability at most $\varepsilon$, arbitrary history dependence, a free public random tape, and at most $H$ bits of persistent state, satisfies \[ H\ge \bigl(\log_2(1/\varepsilon)+a_\varepsilon^{\rm c}\bigr)n-o(n), \] under only $u/n\to\infty$. The constant $a_\varepsilon^{\rm c}$ is an explicit variational threshold obtained by preserving the dependence between the parent accepted mass and the successor reservoir. The structural step is a common-continuation transport lemma. A joint posterior KL bound gives a branch-specific survivor support; the same legal delete–insert word transports that support to one successor state, forcing an accepted reservoir. We then keep the parent outside mass $1-X$ in the conditional-entropy argument instead of replacing it by $1-\varepsilon$. This yields a two-variable analytic envelope, with no selected thresholds, dyadic witnesses, or numerical assumptions.

In his study of Cantor-type intersection theorems and fixed points of almost affine mappings, Jachymski \cite{Jachymski} posed four open questions. In this note, we provide an affirmative answer to the fourth question.

Let $(M,\omega)$ be a compact Kähler manifold with its scalar curvature $S(\omega)$, Assume that $M$ has complex dimension at least $3$ and contains a complex submanifold $X$ of complex codimension at least $2$. Let $\sigma: Bl_{X} M \rightarrow M$ denote the blow-up of $M$ along $X$. We show that $Bl_{X} M$ admits a sequence of Kähler metrics $\{\widetilde{\omega}_{i}\}_{i \geq 1}$ whose scalar curvatures $S(\widetilde{\omega}_i)$ converge to $\sigma^{\ast} (S(\omega))$ in the $C^0(Bl_{X} M)$ norm. Our work is motivated by a recent result of Brown, who established the corresponding result for blow-ups at a point. The proof is based on the gluing method for constructing extremal Kähler metrics on blow-ups, together with new analytic tools and several modifications needed in our setting.

A multisection (as defined by Ben Aribi, Courte, Golla and Moussard) is a decomposition of a closed orientable manifold into model pieces. These model pieces are top-dimensional 1-handlebodies, whose subcollections intersect along 1-handlebodies of lower dimensions, and whose global intersection is a closed surface. This concept extends the notions of Heegaard splittings and trisections to higher dimensions. In this article, we adapt multisections to compact manifolds with boundary, generalizing sutured Heegaard splittings and relative trisections to every dimension. We define the associated diagrams, which encompass sutured Heegaard diagrams and relative trisection diagrams. A relative multisection induces a particular decomposition of the boundary of the manifold, which we call a relative fibration. We show that, in dimension n greater than 3, a connected manifold which relatively fibers is necessarily either a sphere, or a connected sum of copies of the product of the circle with the sphere of dimension n-1. We prove that a compact 5-manifold whose boundary relatively fibers admits a relative multisection. We also state a gluing theorem that allows to combine suitable relatively multisected manifolds with boundary into multisected closed manifolds.

The low-rank alternating direction implicit (ADI) method is an efficient solver for large-scale Stein equations with low-rank solutions. This paper shows that, as in the continuous-time Lyapunov equation case, the low-rank Cholesky factor ADI (LRCF-ADI) method for Stein equations implicitly performs $\mathcal{H}_2$-pseudo-optimal model order reduction for discrete-time systems. This observation leads to an automatic shift-generation strategy, allowing LRCF-ADI to select subsequent shifts without user intervention. The standard LRCF-ADI method requires shifts outside the unit circle. We generalize the method to allow shifts anywhere in the complex plane, including on the unit circle. This extension enables numerical integration for frequency-limited Stein equations by interpolating the integrand at points on the unit circle. It also enables non-intrusive, data-driven balanced truncation and frequency-limited balanced truncation using experimentally measurable transfer function samples on the unit circle, without requiring access to a state-space realization. Numerical results for large-scale models demonstrate the effectiveness of the proposed methods as low-rank Stein equation solvers and data-driven model order reduction methods.

In this paper we establish a criterion on weak type $(1,\,1)$ bound of the following rough singular integral $$T_{\Omega,K,a}f(x)={\rm p.v.}\int_{\mathbb{R}^n}\Omega(x-y)K(x,y)m_{x,y}a f(y)dy,$$ where $m_{x,y}a=\int_0^1a(sx+(1-s)y)ds$ with $a\in L^1(\mathbb{R}^n)$ and $\hat{a}\in L^1(\mathbb{R}^n)$, $\Omega$ is homogeneous of degree zero, integrable in $\mathbb{S}^{n-1}$ and satisfies the cancellation condition $\int_{\mathbb{S}^{n-1}}\Omega(\theta)d\sigma(\theta)=0$ and $K$ is a measurable function defined on $\mathbb{R}^n\times\mathbb{R}^n\setminus \{(x,x):x\in\mathbb{R}^n\}$ and satisfies a Hölder condition. By assuming that $\Omega\in L\log L(\mathbb{S}^{n-1})$ and the operator $T_{\Omega, K}f(x)={\rm p.v.}\int_{\mathbb{R}^n}\Omega(x-y)K(x,y)f(y)dy$ is bounded on $L^2(\mathbb{R}^n)$, we prove the weak type (1,1) bound of $T_{\Omega,K,a}$. As several applications, we obtain a large class of singular integral operators which possess weak type $(1,\,1)$ bound. The main results of this paper essentially extend and generalize some known ones.

Let $F_n$ be the free group of rank $n\geq2$ with basis $x_1,\ldots,x_n$. For $1\leq j<i\leq n$, let $d_{i,j}$ and $e_{i,j}$ be the automorphisms of $F_n$ defined by $d_{i,j}(x_i)=x_ix_j$ and $e_{i,j}(x_i)=x_jx_i$, respectively, and fixing the remaining free generators. Write $D_n=\langle d_{i,j}\mid 1\leq j<i\leq n\rangle$ and $A_n^+=\langle d_{i,j},e_{i,j}\mid 1\leq j<i\leq n\rangle$. We prove that $D_n$ admits a presentation on the generators $d_{r+1,r}$, $1\leq r\leq n-1$, with three families of relations given by commutators of weights two, three, and four, respectively. We extend this to a presentation of $A_n^+$ on the generators $d_{r+1,r}$ and $e_{r+1,r}$, $1\leq r\leq n-1$. These generating sets have minimum cardinality. Moreover, the presentation of $A_n^+$ may be chosen so that every defining relator is a single commutator.

We prove the existence of infinitely many lemniscate-like, butterfly-like, and circular-wave critical points for the length-penalised ideal energy. For the lemniscate and butterfly family, we use a staged direct minimisation process to establish existence. A boundary-layer analysis reveals critical points near two Fresnel phases: $3\pi/4$ modulo $2\pi$, corresponding to lemniscates, and $7\pi/4$ modulo $2\pi$, which are the butterflies. The family of circular waves bifurcates, in a sense, from multiply-covered circles. We use an adapted shooting method to establish their existence.

We study the $X$-ADM mass on complete, one-ended asymptotically flat Riemannian manifolds without boundary in arbitrary dimension $n\geqslant 3$. Starting from a weighted gradient–divergence-free decomposition of the vector field $X$, we construct a conformal metric whose scalar curvature is governed by the critical modified scalar curvature $\mathrm{R}_{X}^{(1-n)}$ associated with $X$. This yields an exact decomposition of the $X$-ADM mass into the ADM mass of the conformal metric plus a nonnegative defect term, which vanishes precisely when $X$ is a gradient, thereby giving an alternative decomposition-based proof of positivity with a full rigidity statement. Explicit negative-mass examples show that the range of parameters $k$ in the modified scalar curvature $\mathrm{R}_{X}^{(k)}$ condition is sharp. We then apply the $X$-positive mass theorem to systems of vector fields, obtaining higher-dimensional multi-charge mass inequalities with rigidity and global alignment in the equality case. This yields the classical three-dimensional electric-magnetic inequality and its higher-dimensional analogue. In higher dimensions, the magnetic datum $\beta$ is a $2$-form and does not have a canonical scalar charge on the asymptotically flat end. We prove that a selected tensorial flux vanishes when $d\beta \in L^1$. Under a prescribed critical asymptotic ansatz, this flux defines a non-trivial tensorial magnetic charge and we formulate a conditional reduction to the $X$-positive mass theorem.

We study the relation between the Gorenstein homological and the Gorenstein cohomological dimension of groups. We prove that for every module of type $FP_\infty$ over any ring the Gorenstein projective and the Gorenstein flat dimensions are always equal to each other. In particular, we have an equality $\mathrm{Gcd}_kG=\mathrm{Ghd}_kG$ for every group $G$ of type $FP_\infty$ over a commutative coefficient ring $k$. For non-locally-finite groups in Kropholler's class ${\scriptstyle\mathbf{LH}}\mathfrak F$, we characterize the groups of Gorenstein (co)homological dimension one, in terms of actions on trees with finite stabilizers. The Hirsch length of a virtually soluble group $G$ determines its Gorenstein (co)homological dimension, even when $G$ has torsion: We show that $\mathrm{Ghd}_\mathbb{Z}G =h(G)$ and, if $G$ is countable, $h(G)\leq\mathrm{Gcd}_\mathbb{Z}G\leq h(G)+1$. Some consequences for elementary amenable groups of finite Hirsch length are also obtained. Finally, we investigate the Gorenstein dimension of modules over certain algebras of groups with torsion and discuss applications to classifying spaces for proper actions.

In this paper we present a generalization of Jordan pairs, called Jordan pair disystem, which appear naturally as the wings of a Leibniz algebra with a finite $\mathbb{Z}$-grading; conversely, we give the Tits-Kantor-Koecher construction for Jordan pair disystems, obtaining a Leibniz algebra with a short $\mathbb{Z}$-grading. This construction extends the construction given by Gubarev and Kolesnikov for Jordan dialgebras (arXiv:0907.1740). We introduce homotopes for Jordan pair disystems at elements and we show that we obtain again a Jordan dialgebra. Moreover, we introduce abelian inner ideals and their kernels for Leibniz algebras, and prove that the subquotient of a Leibniz algebra with respect to an abelian inner ideal is a Jordan pair disystem. This notion of subquotient extends the construction of Jordan dialgebras at $Q$-Jordan elements given by Felipe and Velásquez.

As opposed to classical converse Lyapunov theorems, finite-step converse results are constructive and offer a different starting point: for sufficiently large finite step ahead, say $M$, in an explicit sense, any scaled norm can serve as a converse finite-step Lyapunov function. As for interconnected discrete-time systems, similar lines of argument lead to “non-conservative” small-gain conditions. Motivated by this viewpoint, this paper develops a distributed model predictive control framework for constrained interconnected nonlinear discrete-time systems. Each subsystem solves one local optimization problem at each system time step instant by setting the local stage function in form of a local control finite-step like Lyapunov function, using time-aligned neighbor predictions, optimized state-constraint tightening radii, and a finite-step small-gain terminal inequality. Since received neighbor predictions need not equal the trajectories generated by future receding-horizon optimizations, the nominal converse certificates do not alone ensure recursive feasibility or stability. We therefore develop shift-compatible constraint margins, a local one-step terminal feasibility test for networks that are affine in control, and an analytical bound for the prediction and reoptimization mismatch. The resulting analysis gives recursive feasibility, constraint satisfaction, and a practical $M$-step Lyapunov estimate, with asymptotic convergence when the prediction and reoptimization mismatch bound tends to zero. For constrained linear networks, the conditions reduce to finite-dimensional matrix, QP, and SOCP tests. The framework is specialized to current sharing and terminal bus voltage safety under DC/DC power converters' operational constraints in a two-DGU DC microgrid evaluated on a small laboratory-scale prototype.

In this paper, we investigate the asymptotic behavior of a coupled fluid–rigid body system, consisting of a rigid body immersed in a three-dimensional viscous incompressible fluid, as the body size and the fluid viscosity vanish. We prove that the limiting velocity field is governed by the whole-space incompressible Euler equations. Moreover, the influence of the fluid on the rigid body motion is shown to vanish in this limit.

In this paper, we introduce the notions of extended $\mathcal{O}$-operators on perm algebras, as generalization of $\mathcal{O}$-operators, and the extended perm Yang-Baxter equation. We demonstrate that an $\mathcal{O}$-operator of weight $\lambda$ on a perm algebra gives rise to a post-perm algebra, and an extended $\mathcal{O}$-operator yields a new perm algebra structure. We also give an equivalent characterization of extended $\mathcal{O}$-operators in terms of ordinary $\mathcal{O}$-operators via the $\pi_\pm$ decomposition. The notion of the generalized perm Yang-Baxter equations is introduced, and their relationship with extended $\mathcal{O}$-operators is established. The tensor form of extended $\mathcal{O}$-operators leads to the notion of the extended perm Yang-Baxter equation (extended perm-YBE), which generalizes the perm Yang-Baxter equation. We establish that a solution of the extended perm-YBE with $(R, \ad)$-invariant skew-symmetric part is characterized by an extended $\mathcal{O}$-operator. Furthermore, relationships among extended $\mathcal{O}$-operators, the perm Yang-Baxter equation and the extended perm-YBE are investigated through the framework of quadratic perm algebras and semi-direct product perm algebras.

Motivated by the classical Lagrange spectrum and the reciprocal formulation of the Lagrange spectrum in continued-fraction theory, we introduce a normalized approximation function $\mathcal{N}(\alpha)$ for approximation by multiple zeta-star values. At each depth, the approximation error is minimized over all admissible indices and normalized by the binary scale determined by their weights. The function $\mathcal{N}(\alpha)$ is then obtained by taking the limit inferior as the depth tends to infinity. Using the order structure of multiple zeta-star values, we establish the regularity and generic behavior of this function. We prove that the depthwise approximation functions are upper semicontinuous, that $\mathcal{N}$ is Borel measurable, and that its zero set is a dense $G_\delta$ subset of $(1,+\infty)$. We also derive a natural-prefix approximation estimate showing that a large next digit produces an exceptionally good normalized approximation. Combined with the author's metric resultson Diophantine approximation of multiple zeta-star values, this gives a full-measure theorem under a logarithmically reinforced divergence condition and, in particular, proves that $\mathcal{N}(\alpha)=0$ for Lebesgue almost every $\alpha>1$. Finally, we give some basic properties of the image of the normalized approximation function and show that the image $\mathrm{Im}(\mathcal{N})$ is dense on the extended half-line $[0,+\infty]$.

We consider the open XXZ spin chain with unparallel boundary fields, in the special case in which the Vertex-IRF transform of the K-matrix labelling the boundary field at site 1 is diagonal and invariant under the value of the gauge parameters, the other K-matrix being arbitrary. We discuss here several properties related to this special gauge invariant boundary condition. Such a model can be solved in the Vertex-IRF framework with arbitrary gauge parameters: we can diagonalise the transfer matrix by means of generalised Sklyanin's Separation of Variable (SoV) approach in a family of SoV bases that depend on two arbitrary gauge parameters $\alpha$ and $\beta$, $\beta$ being the so-called dynamical parameter and $\alpha$ an arbitrary shift of the spectral parameter. We show here that such SoV bases depend actually on $\alpha$ and $\beta$ through the difference $\alpha-\beta$ only. Considering the ungauged limit in which $\alpha-\beta$ tends to infinity, we show how to compute all elementary building blocks for the correlation functions of this model. We finally show how to compute overlaps between eigenstates of this model and eigenstates of a spin chain sharing the same boundary condition at site $N$ and in which the gauge invariant boundary condition at site 1 has been changed to a more general one.

Interacting Bose–Hubbard dynamics is formulated on the concrete Buchholz algebra in Fock space. Under repulsive finite-range interactions and a low-activity gap, a discrete functional integral gives volume-uniform exponential local-number bounds at all temperatures. These bounds remove the local-moment hypothesis and yield finite-volume approximation and KMS accumulation states in arXiv:2505.13170.

We introduce MCSI, a two message key exchange we design over the CSIDH class group action, in which each party sends its ephemeral public element under an authenticated encryption keyed by the value the two static keys determine. The design gives implicit mutual authentication, hides the ephemeral element from an eavesdropper, and lets a recipient discard an unauthenticated message after one tag check rather than after an evaluation of the group action, which is four orders of magnitude more expensive. On the analytic side, we prove that our protocol is correct with zero error, and prove three statements in the random oracle model, all reducing to the strong parallelisation problem: indistinguishability of the session key against a passive adversary, confidentiality of the blinded ephemeral element, and integrity of the blinded transport. None uses the decisional group action assumption, which is false for class group actions of non-prime discriminant. We also show that a blinding key cannot come from the session secret it is meant to establish. To instantiate the design we select parameters and show that a prime chosen for elliptic curve discrete logarithms is unusable: for $p = 2^{521}-1$, the NIST P-521 prime, the action admits no efficiently evaluable generator. On the practical side, we build and test the design. We implement the protocol twice, in C and independently in Python, cross check the two, and measure what a session costs in field operations, time and memory. We also audit our code for secret dependent control flow: the field arithmetic and the symmetric layer show none, while the group action leaks the key by construction, and two hundred timings separate two keys whose one-norms differ by five out of 370. Finally, we state what we do not prove, among them security under ephemeral key reveal, forward secrecy of the blinding, and constant time execution.

This work presents a mathematical and computational framework for computing the dispersion relations of shear horizontal (SH) waves in continuously varying heterogeneous layered structures. The approach isolates the contribution of the heterogeneous layer from the complete dispersion relation, learns this contribution using a physics-informed neural network (PINN) and subsequently incorporates the trained model to determine the complete dispersion relation. The mathematical properties of the discretized problem are investigated, including the singularity of the finite-difference system and the oscillatory behavior of the layer solution, while a generalization-error estimate is established for the PINN approximation. The framework is first tested on a seismological configuration consisting of a heterogeneous sandstone layer over a granite half-space, where exponential heterogeneity is considered with independent variation rates in shear modulus and density. The proposed approach is validated against analytical solutions in special cases, while for general configurations the Haskell matrix method demonstrates convergence toward the continuously varying dispersion relation predicted by the PINN as the number of homogeneous sublayers increases. Parametric studies further confirm consistency with the underlying physics. An important feature of the method is that the heterogeneous-layer equation can be trained independently and reused in multiple settings. To demonstrate this, the same trained network is coupled with piezoelectric and piezomagnetic substrates governed by fundamentally different physical laws, highlighting the potential of the PINN framework as a reusable computational module for dispersion analysis in heterogeneous layered media.

A threshold graph is a graph that can be constructed from the one-vertex graph by repeatedly adding either a dominating vertex or an isolated vertex. Motivated by an induced Ramsey-type problem for this class, we define $r'_2(s)$ to be the minimum integer $n$ such that every $n$-vertex graph contains an induced threshold graph on $s$ vertices. We establish exponential upper and lower bounds for $r'_2(s)$ and determine its exact values for $s\in\{3,4,5,6\}$. To study this problem from an edge-coloring perspective, we use the notion of an orderable coloring, introduced by Richer [J. Combin. Theory Ser. B, 80(1) (2000), 172–177]. An edge-colored graph is orderable if its vertices can be ordered so that, for each vertex, all edges from it to later vertices have the same color. Equivalently, $r'_2(s)$ is the minimum $n$ such that every $2$-edge-coloring of $K_n$ contains an orderable $K_s$. We also determine the exact value of the unordered canonical Ramsey number $CR(s, 3)$ for all $s \ge 3$, where $CR(s,3)$ denotes the minimum integer $n$ such that every edge-coloring of $K_n$ contains either an orderable $K_s$ or a rainbow $K_3$. More generally, for graphs $G$ and $H$, we study $r'_2(G)$, the corresponding $2$-color Ramsey number for an orderable $G$, and $CR(G,H)$, where the alternative is a rainbow $H$. For complete bipartite graphs, we prove that for every fixed $s$, $r'_2(K_{s,t}) = CR(K_{s,t}, K_3)= \left(\frac{2^s}{s+1}+o(1)\right)t$ as $t\to\infty$. For $s\in \{2,3\}$, we further determine the exact values of these parameters for infinitely many $t$, using constructions arising from strongly regular graphs, Hadamard matrices and conference matrices.

The Erdős-Pósa theorem states that for every integer $k\geq1$, every graph contains either $k$ vertex-disjoint cycles or a set of $\mathcal{O}(k\log k)$ vertices meeting all cycles. This fundamental min-max duality has been extended to numerous settings, including long cycles, $S$-cycles, that is, cycles containing a vertex in a prescribed set $S$, and cycles satisfying various additional constraints. In contrast, much less is known when the packing itself is required to be induced, namely, when distinct cycles are vertex-disjoint and have no edges between them. We prove that long $S$-cycles admit an induced version of the Erdős-Pósa-type duality. More precisely, we show that there exists a polynomial function $f(k,\ell)$ such that for all integers $k\geq1$ and $\ell\geq3$, every graph contains either an induced packing of $k$ $S$-cycles of length at least $\ell$ or a set of at most $f(k,\ell)$ vertices whose closed neighbourhood intersects all $S$-cycles of length at least $\ell$. The proof introduces a new ear-decomposition technique based on fragile ears and yields a polynomial-time algorithm for every fixed $\ell$.

We prove an almost-everywhere Fourier central limit theorem for stationary random measures on $\bR^d$ with local second moments whose translation action is essentially free and has completely positive entropy. For the resulting almost-everywhere defined Bartlett density $s_\eta$, we show that there is a single $\lambda_d$-conull set of frequencies, independent of the test functions, on which finite collections of normalized smooth-window Fourier transforms converge jointly to proper complex Gaussian limits with covariance determined by $s_\eta$. No quantitative mixing, correlation-decay, or cumulant-summability assumption is imposed. For stationary point processes of positive intensity, the same good-frequency set yields Gaussian limits for ball-window Fourier transforms and exponential limits for their squared moduli. We also construct a stationary ergodic zero-entropy random measure with bounded continuous Bartlett density, positive $\lambda_1$-almost everywhere, for which the Fourier central limit theorem fails.

We construct Wakimoto-type free field realizations of $N=1$ SUSY affine vertex algebras associated with basic classical Lie superalgebras. Using the superfield formalism, we obtain a SUSY analogue of the Wakimoto realization in terms of $\beta\gamma$–$bc$ systems and boson–fermion systems (or SUSY Heisenberg vertex algebras). We also construct screening operators for the free field realizations. A distinctive feature of the SUSY setting is that, unlike the ordinary Wakimoto realization, the injectivity of the free field homomorphism depends on the level. We prove that the homomorphism is injective for nondegenerate $\kappa$, whereas at $\kappa=0$ it fails to be injective. Thus, the SUSY Wakimoto realization exhibits a degeneration at $\kappa=0$ that has no direct counterpart in the ordinary affine case.

We use the relative Abel–Jacobi construction of Cheltsov–Loginov–Orlov–Prokhorov to study unirational Fano threefolds. Our main result is a $50$-dimensional family of smooth unirational del Pezzo threefolds of degree one. We also construct explicit one-parameter families of terminal unirational sextic double solids and quartic threefolds of Picard rank one. Finally, we reinterpret the Roth–Iskovskikh unirationality construction for smooth double quadric threefolds in terms of a correspondence induced by the relative Abel–Jacobi map.

We prove that every Steklov isospectral family of compact smooth planar domains is compact in the $C^\infty$ topology, answering an open question of Colbois, Girouard, Gordon, and Sher. The proof has two main parts. First, a trace comparison principle for Dirichlet-to-Neumann operators yields monotonicity of negative Steklov zeta values and compactness within a fixed conformal class for genus-zero flat surfaces. Second, we analyze the normalized Steklov determinant on degenerating hyperbolic surfaces with geodesic boundary. Its asymptotics are expressed in terms of shrinking boundary components and small Neumann and Dirichlet eigenvalues, which in genus zero are compared with weighted graph Laplacians. This rules out degeneration under a bound on the total hyperbolic boundary length. We finally establish this bound for planar domains by geometric non-collapse estimates.

We prove that two simple parabolic rational maps with conformally conjugate completely invariant parabolic Fatou domains are globally conformally conjugate. The proof combines quasiconformal rigidity with the absence of invariant line fields for simple parabolic maps. Together with our previous existence theorem, the rigidity result gives existence and uniqueness of simple parabolic models for infinitely-connected fixed parabolic Fatou domains. We further obtain a rational-map analogue of the Goldberg–Milnor perturbation conjecture.

We identify, functorially, the boundary limit of a simply defined presentation functor with the dimension quotient, both for groups and for Lie rings over integers.