Papers

Evidential argumentation extends Dung's abstract argumentation by requiring arguments and interactions to be backed by chains of evidence rooted in prima-facie elements. However, existing formalisms lack a unified treatment of evidential support, higher-order relations (attacks and supports targeting arbitrary elements), and collective interactions (sources as sets). In this paper, we introduce the Evidential-Based Higher-Order Set Argumentation Framework (EHSAF), which conservatively generalises several existing frameworks within a single expressive setting. We develop two complete semantics for EHSAFs: an adjacent complete labelling semantics that admits multiple truth values (true, false, undecided) for arguments in support cycles, reflecting an open epistemic attitude toward future evidence; and an extension-based complete semantics that follows a strict evidentialist stance, accepting only arguments with well-founded support chains. We show that these two semantics diverge in the presence of support cycles, and prove their equivalence under support-acyclicity. To enable computational reasoning, we provide a normal propositional encoding of EHSAFs and prove that, in three-valued Łukasiewicz logic, its models correspond precisely to the adjacent complete labellings. We further extend this encoding to continuous fuzzy logics (Gödel, Product, and Łukasiewicz), defining a continuous fuzzy normal encoded semantics. We establish that this fuzzy semantics satisfies key properties—continuity, monotonicity, boundary conditions, and solution existence—and that its ternarisation recovers the adjacent complete labellings under natural t-norm conditions. Our framework thus unifies expressive argumentation with principled three-valued and fuzzy semantics, bridging the gap between qualitative and quantitative reasoning about evidence.

Perturbed plane waves are fundamental objects in scattering theory on Euclidean space and asymptotically Euclidean spaces. In this paper, we investigate the structure of perturbed plane waves in the $\textit{forward}$ direction, in which the outgoing spherical wave is typically singular and conjoined to the incoming plane wave. Melrose & Zworski provided a microlocal description (in the more general setting of asymptotically conic manifolds) using their notion of Lagrangian distributions associated to pairs of intersecting Legendrian submanifolds, on the way to proving that the S-matrix is an FIO. Here, we revisit the problem in the asymptotically Euclidean case, for which the oscillatory integrals used by Melrose–Zworski attain their most complicated form (relative to the more general asymptotically conic case). We seek a more elementary description in terms of physical-space asymptotics. These are specified using a two-faced compactification $X\hookleftarrow \mathbb{R}^d$, with one face for each asymptotic regime. We prove full polyhomogeneity. A transport equation arises as a model problem at the main face (`bf'). The quantum inverted harmonic oscillator arises as a model problem at the front face (`ff').

We prove that two invariants of smooth 4-manifolds defined in terms of trisections are uncomputable: the trisection genus and the Kirby-Thompson L-invariant. That is, there does not exist an algorithm that takes as input a triangulated closed orientable 4-manifold and outputs one of these quantities. The result on the L-invariant resolves the second part of Problem 4.116 in the K3 problem list. In the same spirit, we show that the PL multisection genus of PL manifolds is uncomputable in dimensions at least four.

In 2009, Peterzil and Starchenko proved the following beautiful generalization of Chow's theorem: An entire complex analytic set $X\subset \mathbb{C}^n$ that is definable in an o-minimal structure on $\mathbb{R}$ must be an algebraic set. This result is known nowadays as the o-minimal Chow theorem. In this article, we present some geometric and metric versions of Chow's theorem that generalize the o-minimal Chow's theorem. For instance, we prove that if $X\subset \mathbb{C}^n$ is a pure $d$-dimensional entire complex analytic set, then the following statements are equivalent: (1) $X$ is a complex algebraic set; (2) $\mathcal{H}^{2d+1}(C(X,\infty))=0$, where $\mathcal{H}^{k}(A)$ denotes the $k$-dimensional Hausdorff measure of $A$, and $C(X,\infty)$ denotes the tangent cone at infinity of $X$; (3) $\mathcal{H}^{2d+2}(C_{\mathbb{C}}(X,\infty))=0$, where $C_{\mathbb{C}}(X,\infty)$ denotes the complex tangent cone at infinity of $X$; (4) $\mathcal{H}^{2d}(Z(X,\infty))=0$, where for $A\subset \mathbb{C}^k$, $Z(A,\infty)=\{[v]\in \mathbb{C}P^{k-1};v\in C_{\mathbb{C}}(A,\infty)\}$; (5) For any $k\in\{d+1,...,n\}$ and for any projection $\pi\colon \mathbb{C}^{n}\to \mathbb{C}^{k}$ such that $\pi^{-1}(0)\cap C_{\mathbb{C}}(X,\infty)=\{0\}$ and $Y=\pi(X)$, $\mathcal{H}^{2d}(Z(Y,\infty))<\mathcal{H}^{2d}(\mathbb{C}P^{d})$.

We determine the Grundy total domination number and the skew zero forcing number for every Cartesian product of two paths or cycles, thereby closing the nonmatching bounds previously known for these rectangular, cylindrical, and toroidal families. For $2\le a\le b$, \[ Z_-(P_a\square P_b) =a-\mathbf1_{\{a\ {\rm odd},\,b\ {\rm even}\}}. \] For $p\ge2$ and $c\ge3$, \[ Z_-(P_p\square C_c)= \begin{cases} \min\{p,c\},&c\text{ is odd}, \min\{2p,c\},&c\text{ is even}. \end{cases} \] Finally, for $3\le a\le b$, \[ Z_-(C_a\square C_b)= \begin{cases} 2a-1,&a=b\text{ odd}, 2a,&a=b\text{ even}, \min\{b,2a\},&a<b,\ a\text{ odd}, a,&a<b,\ a\text{ even},\ b\text{ odd}, 2a,&a<b,\ a,b\text{ even}. \end{cases} \] In every case, $\gamma_{\mathrm{gr}}^t(G)=|V(G)|-Z_-(G)$. The path-containing lower bounds use Kronecker differences of skew-symmetric or hollow symmetric factor matrices, and they also determine maximum skew nullity and minimum skew rank. They further produce an infinite family for which maximum skew nullity is strictly smaller than skew zero forcing. The new cycle–cycle lower bounds use a cyclic column-defect estimate, applied to complete columns when the shorter cycle is odd and, after a one-sided bipartite reduction, to half-columns when both cycles are even. Explicit forcing constructions give matching upper bounds throughout.

The multiplicative persistence problem studies the process of repeatedly replacing a positive integer by the product of its digits until a single digit, called the terminal digit, is reached. The classical conjecture asserts that no decimal integer requires more than \(11\) iterations. Brier, Clavier, Gutsche, and Naccache proved the conjecture for all odd terminal digits. In their approach to nonzero even terminal digits, they were led to infinite families of decimal integers in which the numbers of digits \(2,\ldots,9\) are fixed, while arbitrarily many digits \(1\) may be inserted. They conjectured that, despite the infinitude of such a family, the exponent of \(2\) dividing its elements is uniformly bounded. We prove this conjecture and obtain an explicit bound depending only on the prescribed digit multiplicities. More generally, our argument applies in every base \(b\geq3\) and to every prime \(p\mid b\). In base \(10\), combining our bound with the method of Brier, Clavier, Gutsche, and Naccache yields a finite algorithm for a further analysis of each nonzero even terminal digit.

We study the bifurcation of two-dimensional Darcy-Bénard convection (DBC) in a rectangular domain, a canonical model for thermal convection in porous media with applications in geophysics and engineering. The momentum equation lacks advection and viscous dissipation, being regularized solely by a linear Darcy damping term. As a result, the linearized operator generates a semigroup that is neither analytic nor compact and the nonlinear term fails to be Lipschitz. The system is thus placed outside the scope of the standard center-manifold theorem. To overcome these obstructions, we develop a center-manifold reduction adapted to DBC system. Our main result is a constructive proof of the existence of the center manifold function h and local attractivity of the center manifold-the exponential convergence of small solutions toward it-without relying on analyticity of the full linear semigroup and Lipschitz nonlinearity. We circumvent these difficulties by exploiting the partially dissipative structure: the temperature equation is governed by the Laplacian, which generates an analytic semigroup and provides the smoothing needed to compensate for the lack of regularity in the velocity and the absence of global Lipschitz bounds. Through carefully designed inequalities, we establish both the construction of the center manifold function h and the exponential convergence of nearby solutions. Explicit approximations for the center manifold are derived in two scenarios-one simple eigenvalue and two distinct eigenvalues-yielding reduced systems of ordinary differential equations whose analysis determines the bifurcation type. Numerical simulations are presented to corroborate the theoretical results.

In this paper, we establish an analytical model for a class of hyponormal contractions $T$ satisfying $Ran[T^*,T] \subseteq \mathcal D_{T^*}\ominus \mathcal D_T$, where $\mathcal D_{T^*}$ and $\mathcal D_T$ are the associated defect spaces for $T$. We also show that, under a certain condition, a contraction $T$ with $\mathcal D_T \subseteq \mathcal D_{T^*}$ is hyponormal.

Let $\mathbb{P}_n$ be the cone of $n\times n$ complex positive definite matrices. For $0<\theta<1$ and $r\ge0$, let $M_r^{(\theta)}$ denote the weighted matrix power mean, with the logarithmic endpoint at $r=0$, defined by $$ M_r^{(\theta)}(A,B) = \begin{cases} \big((1-\theta)A^r+\theta B^r\big)^{1/r}, & r>0,\\[2mm] \exp\big((1-\theta)\log A+\theta\log B\big), & r=0. \end{cases} $$ For $p>q\ge0$, we introduce $$ \Phi_{p,q}^{(\theta)}(A,B) = \operatorname{Tr}\big[M_p^{(\theta)}(A,B)-M_q^{(\theta)}(A,B)\big]. $$ We prove that $\Phi_{p,q}^{(\theta)}$ is a quantum divergence in the sense of Bhatia-Gaubert-Jain. Moreover, for $$ 0\le q\le1\le p\le2,\qquad p>q, $$ it is jointly convex and satisfies the data processing inequality for completely positive trace-preserving maps, with the output side interpreted by the regularized boundary convention whenever it is singular. The construction recovers Hellinger-type and Jensen-Shannon-type quantities as special cases or limits. We also compute the induced local quadratic form and provide scalar and matrix obstructions showing that the convexity and data-processing range cannot be extended to all parameters.

We determine the complete eigenvalue region of the following Markov chains on $2q$ states, for $q\ge2$. The states are $A_0,\ldots,A_{q-1}$ and $B_0,\ldots,B_{q-1}$. Each transition $A_j\to A_{j+1}$ and $B_j\to B_{j+1}$, for $0\le j<q-1$, has probability one. From $A_{q-1}$, the chain moves to $A_0$ with probability $a$ and to $B_0$ with probability $1-a$. From $B_{q-1}$, it moves to $B_0$ with probability $b$ and to $A_j$ with probability $(1-b)p_j$. Here $a,b\in[0,1]$ and $p_j\ge0$ with $\sum_{j=0}^{q-1}p_j=1$; these parameters vary over all permitted values. Every segment from zero to an attainable eigenvalue lies in the region. Its unit-circle points are exactly the roots of unity of order at most $2q$, as for unrestricted stochastic matrices of the same order, yet the full region is strictly smaller. It excludes $0<|\operatorname{Arg}\lambda|<\pi/q$. For odd $q$, the final nonreal boundary arc ends at a negative real point inside the unit disk; the remaining interval to $-1$ is also boundary. Every transition matrix has the same characteristic polynomial as one with $a=b$. Among $q+1$ fixed matrices in this balanced family, convex combinations of at most three realize every attainable eigenvalue, and two suffice on the boundary. Both bounds are sharp within this set. Consecutive reduced fractions in $[1/(2q),1/2]$ with denominators at most $2q$ select the upper nonreal boundary constructions. Each boundary radius is the unique solution of an equation whose left-hand side increases strictly with the radius.

We formulate a dg-categorical analogue of the Cartan determinant conjecture using numerical Grothendieck groups. Let $D$ be a smooth proper dg category over an algebraically closed field $k$. The Euler pairing on $K_0(D)$ descends to a nondegenerate bilinear form on the numerical Grothendieck group $K_{\mathrm{num}}(D)$, and hence defines a numerical Cartan determinant $\det M_\chi(D)$. While the classical Cartan determinant conjecture predicts that $\det M_\chi(A)=1$ for finite-dimensional smooth algebras $A$, this equality is no longer expected for general smooth proper dg categories. However, we propose the following positivity question: $\det M_\chi(D)>0.$ Our main result proves this positivity when $D=\operatorname{Perf}(X)$ for a large class of smooth projective varieties $X$. More precisely, we show that $\det M_\chi(X)>0$ if either $\dim X$ is odd, or $X$ has even dimension and satisfies the numerical Hodge standard conjecture.

We study the reducibility over $\mathbb{Q}$ of $F_{p,e}(x)=rx^m+p^ef(x)$, where $r\in\mathbb{Z}\setminus\{0\}$, $f\in\mathbb{Z}[x]$, and $0\le m<n:=\mathrm{deg}\, f$, for primes $p$ above explicit coefficient-dependent thresholds. For arbitrary $e\ge1$, we determine the degrees, endpoint $p$-adic valuations, reductions modulo $p$, and heights of all nonconstant proper integral factors. Writing $\delta=\gcd(e,m,n)$, we show that every such factor has degree $tn/\delta$ for some $1\le t\le\delta-1$; in particular, $\delta=1$ implies irreducibility. For $e=2$ and $e=3$, we obtain effective necessary and sufficient criteria in terms of at most two associated binary quadratic forms and at most one associated binary cubic form, respectively. The quadratic criterion reduces to at most two fixed Pell-type equations with prescribed second coordinate $p$. In the cubic case, Thue's theorem yields finiteness of the reducible primes whenever the associated form is irreducible over $\mathbb{Q}$.

We determine the complete ordered solution set of the Fermat-type matrix equation $X^n+Y^n=Z^n$ in $\mathrm{GL}_2(\mathbb{Z})$ for every integer $n\ge3$, expressed in terms of simultaneous integral-conjugacy orbits while keeping the variable symmetries of the equation separate. The equation is solvable if and only if $4\nmid n$ and $6\nmid n$. Thus, in contrast with the determinant-one setting, odd multiples of $3$ are solvable in the full group $\mathrm{GL}_2(\mathbb{Z})$ and necessarily involve determinant-$-1$ components. For every admissible even exponent, the full ordered solution set has a canonical disjoint orbit decomposition indexed by the same transversal that occurs in the matrix Pythagorean equation. For odd exponents with $3\nmid n$, the commuting solutions form exactly six simultaneous-conjugacy orbits. The noncommuting odd solutions form exactly twenty simultaneous-conjugacy orbits when $3\nmid n$, and exactly fourteen when $3\mid n$. Under the coarser equivalence generated by signed variable permutations and simultaneous integral conjugation, these noncommuting orbits collapse to four and three classes, respectively. The proof combines the canonical orbit classification of the matrix Pythagorean equation, Cayley–Hamilton recurrences, commutator divisibility, trace-growth estimates, integral conjugacy of torsion matrices, and an elementary symmetry-reduced trace argument for the cubic equation.

Let $U$ be a connected open subset of the complex unit ball $B^n$, $n\geq2$, and let $F_i: U \to \Omega_i$ be non-constant holomorphic maps into irreducible bounded symmetric domains. We prove that if $F=(F_1, \cdots, F_m)$ is a holomorphic isometric map from $U$ to the product of $\Omega_i$, then every $F_i$ extends to a proper holomorphic isometric embedding up to a positive constant from $B^n$ into $\Omega_i$.

Over an arbitrary field of characteristic zero, we construct a two-dimensional preLie algebra and a two-dimensional postLie algebra that do not embed into the corresponding induced structures of any finite-dimensional preassociative or postassociative algebra, respectively.

We extend orbifold irregular Hodge theory from simplicial toric Landau–Ginzburg models to data defined by possibly non-simplicial stacky fans. For a Clarke mirror pair, we construct a filtered Koszul–de Rham complex over toric face rings. For simplicial ray-supported data, a strict filtered comparison identifies this complex with the Harder–Lee cellular residue complex, and hence it computes the orbifold irregular Hodge numbers. Using this complex, for a fixed possibly non-simplicial datum, we prove that the models obtained from its projective crepant simplicial subdivisions have the same orbifold irregular Hodge polynomial whenever the potential is nondegenerate at infinity. This common value defines the stringy irregular Hodge numbers of the original datum.

We study linear cocycles over ergodic translations of the two-torus acting on Hermitian vector bundles described by unitary sewing matrices. For a continuous invariant complex line subbundle $L$, we prove that $c_1(L) \ne 0$ obstructs every nonzero measurable eigensection under a measurable phase-coboundary condition. The abstract obstruction holds in every rank, while for rank-two analytic cocycles with a dominated splitting over a Diophantine translation the phase condition is automatic. In that regime the criterion is exact: $L$ carries a nonzero measurable eigensection if and only if $c_1(L)=0$ and the winding vector of the multiplier phase vanishes. When these conditions hold, the realized eigenvalues form a dense coset on a circle; for each such eigenvalue, the eigenspace carried by $L$ is one-dimensional and has a real-analytic nowhere-vanishing generator. We also construct a Liouville example showing that the Diophantine hypothesis cannot be removed. The proof converts the Chern number into a magnetic charge on the universal cover and applies covariance for irrational magnetic translations.

We introduce the differential perm bialgebras by generalizing the study of perm bialgebras to the context of differential perm algebras. They are characterized by the Manin triple of differential perm algebras as well as the matched pairs of differential perm algebras. The quasi-triangular (resp. triangular, factorizable) differential perm bialgebra are introduced. In particular, symmetric solutions of an analogue of perm Yang-Baxter equation in differential perm algebras provide triangular differential perm bialgebras, whereas in turn the notions of $\mathcal{O}$-operators of differential perm algebras. As an application, the relationship between differential perm bialgebras and related bialgebra structures is discussed in detail. We show that there is a natural differential infinitesimal bialgebra structure on the tensor product of a differential perm bialgebra and a quadratic Zinbiel algebra, and this differential infinitesimal bialgebra structure is quasi-triangular (resp. triangular, factorizable) if the original differential perm bialgebra is quasi-triangular (resp. triangular, factorizable). Starting from a differential perm bialgebras with certain conditions, we can construct Novikov bialgebras, and these Novikov bialgebras can also be implemented by using the tensor product of diNovikov bialgebras and quadratic Zinbiel algebras.

Let $\mathcal O_{2A_1}$ be the next-to-minimal nilpotent orbit of the complex simple Lie algebra of type $E_7$. Let $\mathfrak p_7$ be the maximal parabolic subalgebra corresponding to the simple root $\alpha_7$, and let $\mathfrak u^+$ be its nilpotent radical. We prove that the affinization $T^*(\mathcal O_{2A_1}\cap\mathfrak u^+)^{\mathrm{aff}}$ of the cotangent bundle of $\mathcal O_{2A_1}\cap\mathfrak u^+$ is isomorphic to the nilpotent orbit closure $\overline{\mathcal O}_{2A_1}$ as affine varieties.

Let $T_r(n)$ denote the complete $r$-partite graph on $n$ vertices whose part sizes differ by at most one. A graph $H$ is called $K_{r+1}$-Turán-good if, for all sufficiently large $n$, the graph $T_r(n)$ contains the maximum number of copies of $H$ among all $n$-vertex $K_{r+1}$-free graphs. We say that $H$ is $K_{r+1}$-Turán-stable if, for every $\varepsilon>0$, there exist $\delta>0$ and $n_0$ such that, whenever $n\ge n_0$ and an $n$-vertex $K_{r+1}$-free graph $G$ contains at least $\mathrm{ex}(n,H,K_{r+1})-\delta n^{v(H)}$ copies of $H$, the edit distance between $G$ and $T_r(n)$ is at most $\varepsilon n^2$, where $v(H)$ denotes the order of $H$. In this paper, we prove that every graph $H$ is both $K_{r+1}$-Turán-good and $K_{r+1}$-Turán-stable for $r\ge 4v(H)^3+11v(H)^2$. This strengthens results of Morrison, Nir, Norin, Rzążewski, and Wesolek~[J. Combin. Theory Ser. B, 2023] and Gerbner and Hama Karim~[J. Graph Theory, 2024], and gives a positive answer to a question of Morrison, Nir, Norin, Rzążewski, and Wesolek. We also prove that $\inj(H,G)\le \inj(H,T_r(n))$ for every $n$-vertex $K_{r+1}$-free graph $G$ and for $r\ge 40v(H)^3$, where $\inj(H,G)$ denotes the number of injective homomorphisms from $H$ to $G$. Finally, we give a negative answer to another question of Morrison, Nir, Norin, Rzążewski, and Wesolek.

We study affine reflection transforms of convex polygons with parallel opposite sides. An opposite-side involution sends a pencil through an interior point to a piecewise smooth family of lines. Its completed envelope consists of conic arcs and transition segments. The completed transform has non-positive oriented area. Maximising the negative oriented area and dividing by the polygonal area gives an affine invariant that vanishes precisely for centrally symmetric polygons. Combining an identity for reflected chord vectors with the Rogers–Shephard inequality shows that this invariant is strictly less than $1$, with optimal universal constant $1$. The energy extends continuously to the boundary and vanishes there. We also compare chord counts under compatible smoothing and prove Hausdorff convergence to the smooth transform.

We characterize all lattices $\Lambda \subset \mathbb{R}^2$ of rational density for which the Gabor system generated by the first Hermite function along $\Lambda$ is a frame for $L^2(\mathbb{R})$. We prove that these are precisely the lattices whose density satisfies $D(\Lambda) = \frac{q}{p} $ where $p,q \in \mathbb{N}$ are coprime with $q \geq p+2$, thereby confirming a conjecture of Lyubarskii and Nes. A formalization of our main result in Lean $4$ is also included.

We establish a noncommutative version of a familiar Johnson–Maurey–Schechtman–Tzafriri Theorem, by showing that for any $0 <p<2$ and a (not necessarily semifinite) von Neumann algebra $\mathcal{M}$ on a separable Hilbert space, if a symmetric quasi-Banach function space $E(0,1) $ containing the function $t\mapsto t^{-1/p}$, $0<t\le 1, $ then there exists a noncommutative probability space $(\mathcal{N},\sigma )$ such that $L_p(\mathcal{M})$ is isometric to a subspace of $E(\mathcal{N},\sigma)$. In particular, this answers two questions raised by Randrianantoanina in 2006.

Multi-Threshold (MT) sampling fixes voltage levels and records when a waveform crosses them. The resulting signal-generated event stream is governed not by a prescribed clock but by the nonuniform geometry of the realized crossing set. Within classical nonuniform sampling, we develop a retained-event MT theory for static, uniformly spaced thresholds. For a realized retained MT time set, Beurling's weak-limit theorem gives the exact necessary-and-sufficient condition for stable sampling of $PW_\Omega$. For a fixed signal class, the same principle gives the a priori criterion: uniform stable MT sampling is equivalent to Bernstein uniqueness for every set in the class weak-limit hull. This criterion reduces to computable forms under structured priors. Periodic templates give finite fiber/rank tests, periodic-gap streams give a Shannon-type complete-interpolation boundary and a stable-sampling density rule, and finite-state gap priors reduce noncritical certification to a maximum-cycle-mean condition. Kadec-type local certificates, retained-density bounds, and finite-matrix tests provide practical sufficient conditions for finite active records, while conditional perturbation bounds quantify how jitter, front-end noise, and threshold error consume a clean sampling margin under event correspondence. The theory also identifies the unavoidable boundary of this model: unrestricted finite-energy retained MT records have zero full-line lower Beurling density and therefore cannot provide a universal prior-free sampling theorem for $PW_\Omega$. Within retained-event point sampling, the MT recovery question is resolved precisely: exact recovery is governed by event-set geometry, computable guarantees require structural priors or finite-dimensional/tail assumptions, and the unrestricted static finite-energy problem is impossible.

After Corollary 1.3 in Calculus of Variations and Partial Differential Equations 60 (2021), Paper 231, Djitte, Fall, and Weth asked whether, when $1<p<2$, a volume-constrained local minimizing domain for the best fractional Sobolev constant must still be a ball. This paper solves that problem. Let $0<s<1$, $1<p<2$, and let $\Omega\subset\mathbb{R}^N$ be a bounded $C^3$ domain. If $\Omega$ is a local minimizing domain for $\lambda_{s,p}(\Omega)=\inf{[u]_s^2:u\in\mathcal{H}*0^s(\Omega),\ |u|*{L^p(\Omega)}=1}$ under smooth volume-preserving deformations, then $\Omega$ is a ball. The proof first uses the fractional Hadamard formula to reduce shape minimality to the overdetermined boundary condition $u/\delta^s=C_0$. To overcome the moving-plane obstruction caused by the failure of $u^{p-1}$ to be Lipschitz at zero, we establish a weighted singular narrow-domain maximum principle whose absorption factor is exactly the $sp/N$ power of the measure of the negative set. The difficulty at a corner is resolved by a finite boundary expansion: setting $\rho=sp$, the boundary quotient is composed of finitely many constant-coefficient normal powers $\delta^{k\rho}$ and a $C^{1,\varepsilon}$ remainder; when $k\rho=1$, the unique resonant correction is $\delta\log\delta$. This expansion makes all lower-order normal terms on the two sides of an orthogonal corner cancel, thereby yielding the first-order tangential vanishing required by the moving-plane corner lemma. The method does not require the domain to be convex and covers the full range $0<s<1$ and $1<p<2$.

Connectivity can turn separate contributions into a public good that benefits participants and nonparticipants alike. Yet only participants pay for it, and each can avoid her charge by withdrawing while enjoying whatever public output survives. How do network connections constrain who can or must pay, and how large a cost can the group bear? We study a participation game in which public output equals the size of the largest connected group of active players, whose members divide a fixed operating cost. The output a participant destroys by withdrawing is her withdrawal responsibility and bounds her payment. These bounds determine the group's cost-bearing capacity and, whenever the group can be kept active, which participants must pay. We prove that among connected networks with at least five players, the path uniquely maximizes the cost-bearing capacity of a group, yet minimizes the expected surviving output after a uniformly selected participant withdraws. As the group grows, the ratio of maximal capacity to gross social benefit under full participation converges to one quarter. Fragility can therefore strengthen voluntary finance, but even the most financeable network cannot close the gap between the value it creates and what its participants will pay to sustain it.

Motivated by the geometric interpretation of spatially homogeneous cosmological models, we formulate a spacetime closedness theorem directly at the Lorentzian level. A classical space-form classification theorem organizes homogeneous and isotropic spatial geometries, but by itself it does not yield a Lorentzian statement about the ambient spacetime. The main technical step is to derive finite slice-volume from genuinely Lorentzian control hypotheses on the foliation. This is achieved in a strong version, for maximally regular globally hyperbolic $(n+1)$–spacetimes with a finite-time Big Bang and homogeneous complete spacelike slices, and in a weaker version in which maximal regularity is replaced by time-integrability of the accumulated expansion rate. In both cases, the argument separates an analytic step, deriving finite slice-volume from the Big Bang and temporal control, from a geometric step, upgrading finite volume to compactness by homogeneity and completeness. The theorem is stated in arbitrary spacetime dimension and is accompanied by a Lean~4 formalization of the strong and weak abstract statements.

Constrained particle trajectories along lower-dimensional world sheets constitute a foundational domain in relativistic kinematics and submanifold theory. This paper establishes a comprehensive geometric and kinematic framework for third-order particle dynamics constrained to timelike surfaces in Minkowski 3-space. By utilising the localised Darboux frame, we explicitly derive the relativistically invariant parameters, namely the geodesic curvature, normal curvature, and geodesic torsion and analyse the non-linear coupling between the intrinsic surface topology and the extrinsic spacetime geometry. Primary focus is dedicated to formulating the relativistic "jerk" vector, which accounts for the instantaneous time rate of change of acceleration and captures the high-order structural stresses of the motion. Furthermore, we generalise Siacci's theorem to this Lorentzian setting, providing a non-perpendicular geometric decomposition of both the acceleration and jerk fields into explicit tangential and radial components relative to a fixed coordinate origin. Our formulations explicitly unveil the hidden central-force dynamics and dimensional compactification mechanisms under planar constraints, showing that out-of-plane torsional shocks vanish identically when the modified geodesic torsion is suppressed. These results offer novel analytical insights for structural stability and conserved angular momentum-like quantities governing constrained physical systems in non-Euclidean spacetimes.

We study long-run optimal intervention strategies for endemic infections in a stochastic susceptible-infected-susceptible (SIS) model. The proportion of infected individuals evolves as a diffusion process with random fluctuations, while a control variable $\zeta_t\in[0,\tilde{\zeta}_{max}]$ represents the intensity of public health interventions that reduce transmission for some intervention threshold $\tilde{\zeta}_{max}\in (0,1]$. The objective is to minimize the long-run average societal cost, balancing the burden of infection against the costs of interventions. Under a concave intervention cost structure the problem can be formulated as an ergodic stochastic control problem, whose structure implies (under certain additional conditions) that optimal interventions are of bang-bang type, switching between no intervention and the maximal admissible intervention at a single switching threshold in the infection level. We construct candidate value functions, rigorously verify optimality in this single-threshold case, and relate the results to extinction and persistence properties of the underlying SIS dynamics in the absence of control. In our framework, the analysis provides a rigorous justification for the threshold-based intervention rules commonly used in epidemic management.

Anticoncentration and entanglement are two properties used to study classical hardness or easiness of Gaussian boson sampling in varying regimes. In this paper, we consider three directions concerning anticoncentration and entanglement in Gaussian boson sampling: (1) We derive closed-form expressions for the second moment of hafnians of symmetric Gaussian products, and use this to precisely locate the anticoncentration transition as a function of the number of squeezed input modes. (2) We derive closed-form expressions for the Rényi-$\alpha$ Page curves for Gaussian boson sampling. (3) We study unequal input squeezing parameters $(s_i)_i$, and prove estimates as well as monotonicity of the average-case Rényi-2 entropy Page curve in terms of the magnitudes $|s_i|$, which allow for extending equal squeezing results to unequal squeezing.