Papers

For a set $X$ of integer points, the relaxation complexity $\operatorname{rc}(X)$ is the smallest number of facets of any polyhedron $P$ such that $P \cap \mathbb{Z}^d = X$. In this paper, we focus on the case where $X$ is the discrete standard simplex $\Delta_d = \{\mathbf{0}, \mathbf{e}_1, \dots, \mathbf{e}_d\}$. We show that $\operatorname{rc}(\Delta_d) = O(\log d)$ by an explicit, elementary construction. This improves upon the previously best-known upper bound $\operatorname{rc}(\Delta_d) = O(d / \sqrt{\log d})$ due to Aprile, Averkov, Di Summa, and Hojny (2024) and matches an asymptotic lower bound by Averkov and Schymura (2022).

In this work we consider tensor train varieties. These are varieties of tensors arising in a range of fields, including quantum many-body physics and machine learning. Using methods from integral geometry, we obtain a combinatorial expression for their degrees. We provide the ready-to-use julia package TTVarietyDegree$.$jl$.$

We revisit the large population limit of our epidemic model with infection age dependent infectivity and progressive immunity waning, under the assumption that the supremum in $t$ of the random infectivity function has a finite expectation, while the previous proofs assumed that this supremum admits a deterministic upper bound.

We study Brauer groups of the smooth loci in linear systems on simply connected smooth projective varieties. Under a suitable ampleness condition, we prove that the Brauer group is at most $\mathbb Z/2 \mathbb Z$. This applies when the underlying variety is the projective plane, a very general K3 surface, or a general cubic fourfold. As an application, we compute the Tate–Shafarevich group parametrizing torsors over the relative Jacobians of universal smooth plane curves. Our approach is via a study of the $2$-nodal locus in the linear system.

Hyperspectral standoff detection systems such as Focal Plane Array (FPA) Fourier Transform Infrared (FTIR) spectrometers provide high spatial resolution in detecting airborne chemical contaminants that are invisible to the human eye but potentially hazardous. When two such systems are operated simultaneously with a suitable opening angle, they enable tomographic reconstruction of contaminant plumes with improved spatial and temporal accuracy. This work presents a mathematical model of these measurement capabilities and an algorithm to identify, localize, and quantify contaminant release sources. The objective is to develop a a tool that reconstructs release locations and predict the future plume evolution from standoff measurement data, thereby supporting early warning and situational awareness in hazardous material release scenarios. The transport of contaminants is modeled by an advection-diffusion equation, and the corresponding inverse problem for source identification is formulated accordingly. Owing to the severe ill-posedness and underdetermination of the problem, a sparsity-promoting regularization approach is employed together with a high-performance optimization algorithm. To incorporate the tomographic measurement data into the discrete formulation, a level-set description of a threshold concentration is used, allowing the measurements to be represented independently of the computational mesh and avoiding costly remeshing procedures.

We prove a sharp singularity theorem in which the classical focusing hypothesis of Hawking–Penrose theory is replaced by a condition on the asymptotic volume growth of timelike geodesics. More precisely, let $(M,g)$ be a smooth, globally hyperbolic Lorentzian manifold satisfying the strong energy condition, and let $V\subset M$ be a smooth, compact Cauchy hypersurface. We associate to the future-directed timelike geodesics orthogonal to $V$ asymptotic volume-expansion invariants $\theta_\alpha$, and show that a uniform lower bound $\theta_\alpha\ge c>0$ forces a sharp upper bound on the time separation between $V$ and its chronological past. In particular, $(M,g)$ is past timelike geodesically incomplete. The estimate is sharp and equality is attained by Lorentzian cones. Moreover, in the smooth setting we prove a rigidity theorem showing that equality characterizes these cone models. The argument extends to globally hyperbolic Lorentzian length spaces satisfying the synthetic strong energy condition in the form of the Timelike Measure Contraction Property, TMCP(0,N). In this non-smooth setting we also obtain corresponding raywise and measure-rigidity statements.

We consider linear programming in the oracle model: $\max\{c^\top x \,:\, x\in P\}$, where the polyhedron $P=\{x\in\mathbb{R}^n\,:\, Ax\le b\}$ is given by a separation oracle. We present an algorithm that finds exact primal and dual solutions using $O(n^2\log(n/\delta))$ oracle calls and $O(n^4\log(n/\delta)+n^5\log\log(1/\delta))$ arithmetic operations, where $\delta$ is a geometric condition number associated with the system $(A,b)$. These bounds do not depend on the cost vector $c$ and do not require a priori knowledge of $\delta$. For rational data, $\log(1/\delta)$ is polynomially bounded in the encoding size of $(A,b)$, thus providing a polynomial-time algorithm. The algorithm works in a black box manner, requiring a subroutine for approximate primal and dual solutions; the above running times are achieved when using the cutting plane method of Jiang, Lee, Song, and Wong (STOC 2020) for this subroutine. Whereas approximate solvers may return primal solutions only, we develop a general framework for extracting dual certificates based on the work of Burrell and Todd (Math. Oper. Res. 1985). Our algorithm strengthens results by Grötschel, Lovász, and Schrijver (Prog. Comb. Opt. 1984), and by Frank and Tardos (Combinatorica 1987) that rely on bit-complexity arguments. Our algorithm avoids rounding-based arguments such as simultaneous Diophantine approximation and uses geometric arguments instead.

Deep Joint Source-Channel Coding (JSCC) has emerged as a promising paradigm for overcoming the “cliff effect" in wireless communications. However, existing Deep JSCC frameworks operate directly on raw analog data such as image pixels rather than the discrete semantic tokens that foundation models require. Moreover, traditional systems employ fixed, hand-designed constellations that treat all tokens equally, leading to catastrophic random errors under channel noise. In this paper, the Semantic Token Codebook Communication (STCC) is proposed as a unified source-channel semantic token coding framework designed to transmit the discrete semantic tokens of foundation models over noisy channels. The core of STCC is the Semantic Token Codec (STC). It accepts discrete tokens as input, which maintains compatibility with foundation models while employing a residual multiple layer perceptron, i.e., MLP-based encoder that learns geometrically structured constellations optimized with a triple-loss objective. This learned mapping forces the channel topology to align with the semantic embedding space, ensuring that channel noise results in topological errors rather than random corruption. This phenomenon is theoretically and empirically characterized, identifying “Semantic Drift" in symbolic modalities and “Structural Distortion" in perceptual modalities, where errors shift predictions to semantically or structurally similar tokens. Extensive experiments demonstrate that STCC significantly outperforms traditional systems in low-SNR regimes, effectively converting channel noise into semantic variations without requiring receiver-side modification.

We prove sub-Riemannian Selberg trace formulae for compact quotients of SL(2,R). Using the Fourier decomposition along the SO(2)-fibers, we reduce the heat trace computation to the Selberg trace formula for Maass Laplacians on the hyperbolic plane. The resulting formula has an identity contribution and a hyperbolic contribution, the latter involving a theta factor that records both the twisting character and the central sign of the chosen lift to SL(2,\mathbb R)). We then use this trace formula to compute the zeta-regularized determinant of the sub-Laplacian. The determinant formula is expressed as a universal factor depending only on the base hyperbolic surface times an explicit lift- and character-dependent Selberg-type product.

This paper investigates the well-posedness issue for a reduced two-dimensional model of micropolar fluids. This reduced model presents a coupling between an Euler-type equation for the velocity field of the fluid and a transport-diffusion equation for the microrotation field (which is a scalar field, in this setting). We establish the local existence and uniqueness of strong solutions in the scale of Besov space $B^s_{p,1}$ having regularity index $s\geq1+2/p$. Furthermore, in the subcritical case when $s>1+2/p$, we prove that these solutions exist globally in time. The global persistence of regularity in the critical setting $s=1+2/p$ remains open.

We propose novel semi-decoupled and fully-decoupled iterative algorithms for efficiently solving the fully-coupled nonlinear four-field thermo-poroelastic model discretized in space by discontinuous Galerkin method on polytopal grids. We present the model problem, its four-field formulation, and the arbitrary-order weighted symmetric interior penalty scheme exploited for its spatial discretization. Such a scheme is robust with respect to strong heterogeneities in the model coefficients. Then, we present the two solution strategies and prove that under suitable conditions both schemes are convergent. A wide set of numerical simulations is presented to assess the convergence and robustness properties of the proposed method. Moreover, we test the scheme with literature and physically sound test cases for proof-of-concept applications in the geophysical context.

Convex optimization problems with linear equality constraints arise ubiquitously in scientific computing, machine learning, and control theory. Classical Krylov methods are effective but rely on problem-specific preconditioners and high memory. Conversely, gradient-based methods like the augmented Lagrangian method (ALM) avoid these issues yet suffer from slow outer iterations. Developing accelerated outer-iteration schemes, therefore, remains a critical research objective. In this study, we demonstrate that incorporating a proximal operation into the augmented Lagrangian framework yields the proximal ALM, where the outer iteration is equivalent to an implicit gradient descent-ascent (GDA) scheme. We further establish that this equivalence extends naturally to the setting of variable step sizes. Through Lyapunov analysis, we show that the underlying potential function must be shifted from the conventional objective gap to a variational inequality measure, signaling a shift in perspective from pure convex optimization to minimax optimization. Motivated by these observations, we first develop an implicit GDA scheme with variable step sizes based on a continuous-time ODE framework, which achieves an $o(1/k)$ last-iterate convergence rate for both the primal-dual objective gap and the gradient norm. Building upon a second-order ODE framework, we then propose a family of Nesterov-type implicit GDA schemes parameterized by $r \geq 0$, which achieves an $o(1/k^{r+1})$ last-iterate convergence rate for the primal-dual objective gap. Furthermore, specializing the second-order ODE formulation to the case $r=0$, we derive a corresponding explicit GDA scheme and prove an $o(1/k)$ last-iterate convergence rate for the primal-dual objective gap. Finally, we present several numerical experiments to validate these theoretical results and demonstrate the effectiveness of the proposed methods.

In this paper, we develop a continuous-time model-free reinforcement learning algorithm to learn deterministic equilibrium policies in general time-inconsistent control problems. Utilizing the extended Hamilton-Jacobi-Bellman system, we recast the original time-inconsistent problem into an equivalent two-stage problem. In the first stage, for given auxiliary functions, we employ the deterministic policy gradient approach to learn an optimal policy in an auxiliary time-consistent control problem. In the second stage, given the updated policy, we exploit the inner fixed point iterations and some martingale characterizations to learn the auxiliary functions. As a theoretical contribution, we provide some mild model assumptions and establish the convergence of inner fixed point iterations. By repeating this actor-critic style of iterations across two stages, our algorithm aims to learn the equilibrium under different sources of time-inconsistency in a unified manner. The superior effectiveness of the proposed algorithm are illustrated in two classical financial applications with time-inconsistency: mean-variance portfolio management and optimal tracking portfolio under non-exponential discounting.

A new Block Coordinate Gradient (BCG) method, dubbed bAdag, for smooth, nonconvex minimization problem is proposed; it falls in the class of Objective Function Free Optimization (OFFO) methods, and it is based on the AdaGrad algorithm. At each iteration, our method computes an adaptive step size based on the cumulative sum of block gradients, instead of full gradients as in AdaGrad-type methods. We prove ergodic, sublinear convergence rates for the bAdag algorithm when minimizing a smooth, possibly nonconvex objective under the (block) Lipschitz continuity assumption on the gradient. Our theory covers three widely popular block selection strategies: the Cyclic (C) rule, Uniform Random selection (UR), and the greedy Gauss-Southwell (GS) rule. We also extend our algorithm and its convergence theory to box-constrained smooth functions. We validate the proposed algorithms through synthetic and real-world experiments.

Let $L(\lambda)$ be a simple highest weight module of a classical Lie algebra $\mathfrak{g}$ with highest weight $\lambda-\rho$, where $\rho$ is half the sum of positive roots. Joseph proved that the associated variety of the annihilator ideal of $L(\lambda)$ (also called the annihilator variety) is the Zariski closure of a nilpotent orbit in $\mathfrak{g}^*$. Recently, Bai–Ma–Wang introduced a partition algorithm to describe this corresponding nilpotent orbit for a given highest weight module $L(\lambda)$. In this paper, we present a new direct proof of Bai–Ma–Wang's partition algorithm using Sommers duality.

We prove that, for a generic $C^m$-domain $\Omega \subset \mathbb{R}^d$ (with $m\geq3$, $d \geq 2$) in the sense of Baire category for the Micheletti topology, every Dirichlet-Laplacian eigenfunction has nonzero average. This gives a standalone spectral-geometric answer to a question raised by Steinerberger and Venkatraman. The proof is based on a direct shape-perturbation argument. A key ingredient, which appears to be of independent interest, is a Baire-category theorem for shapes: domains with real-analytic boundary form a meager subset of the space of $C^m$ domains. It is this input that allows us to bypass, rather than resolve pointwise, the overdetermined elliptic configurations in which the first shape derivative of the mean degenerates. We then derive two consequences. First, from a geometric viewpoint, a generic domain has trivial Euclidean isometry group. Second, in control theory, a generic domain makes the Dirichlet heat equation approximately controllable and rapidly stabilizable, that is, stabilizable at any prescribed exponential decay rate, by means of a single spatially homogeneous scalar control.

We study Kohn-Sham Density Functional Theory (DFT) models describing the electronic structure of two-dimensional materials placed in a three-dimensional environment, encapsulated between two parallel conducting electrodes. In this geometry, the Dirichlet boundary conditions at the electrodes screen the Coulomb interaction, which becomes effectively short-ranged, of Yukawa type. We prove that some nonlinear Kohn-Sham DFT models are well-posed in this setting, both for periodic materials (such as graphene) and for quasi-periodic materials (such as twisted bilayer graphene and other moiré materials for generic incommensurate twist angles).

We establish conditions for embedding a corpus of $N$ documents as $d$-dimensional vectors such that every $k$-subset $S \subseteq [N]$ is realizable as a result of top-$k$ retrieval by some query vector. Recent work shows that $d = O(k)$ suffices for such embeddings to exist in $\mathbb{R}^d$, independently of $N$. We theoretically prove that this corpus-independent bound is specific to infinite precision. With $B$ bits per coordinate, perfect top-$k$ retrieval requires $Bd = \Omega(k \ln N)$; thus, at any fixed precision, the dimension must grow at least logarithmically with $N$. Specializing to a $\ell_2$-normalized $B$-bit uniform scalar quantization model, we also identify a threshold on the precision $B^{*} = O(\ln \ln N)$ below which no dimension suffices, together with two further regimes that bound the feasible $(B, d)$ pairs. Our result implies that in practical vector databases and dense retrieval systems where quantization is standard, the embedding dimension and possibly the precision must grow with the corpus size.

In this work, we settle a problem that dates back to the early 2000s. We provide a complete characterization of special matchings of arbitrary Bruhat intervals in Coxeter groups of type $A$ and apply this result to prove a conjecture of Brenti from 2003 concerning the computation of Kazhdan-Lusztig $R$-polynomials via special matchings. This yields new evidence in support of the Combinatorial Invariance Conjecture.

Multispecies spatial data arise in many applications where interactions between different entities are central to system behaviour, including biomedical imaging, geospatial analysis, and species ecology. Despite their importance, relatively few quantitative tools exist to capture such interactions. In this work, we propose magnitude-based features for the analysis of multispecies spatial data. Magnitude is a real-valued invariant of finite metric spaces that can be interpreted as an effective number of points, incorporating both spatial configuration and scale. We develop global and local magnitude feature vectors and demonstrate their utility on synthetic tumour microenvironment data, and in tissue microarray data from human colorectal cancer samples. Locally, the method identifies distinct neighbourhood types and reveals spatial heterogeneity; in the model, this includes radial patterns associated with different qualitative outcomes of the simulations, while in the real-world data it reflects the importance of tertiary lymphoid structure-like interactions between B and T cell populations. Globally, the approach recovers known classifications of long-term simulation outcomes across parameter regimes in synthetic data, and suggests important roles for CD4+ T cells and CD163+ macrophages in distinguishing patients with favourable Crohn's like reactions from unfavourable diffuse immune infiltration. Together, these results suggest that magnitude-based features provide a powerful and flexible tool for the analysis of multispecies spatial data.

Global nonnegative martingale solutions are shown to exist for a stochastic Keller-Segel system with a nonlocal Fisher-KPP source and super-linear multiplicative noise. The result is obtained for nonnegative initial data with no smallness assumption, provided that the nonlocal source term is dominant. The main difficulty stems from the absence of a coercive structure and the super-linear nature of the noise. An additional cut-off with finite L^2 norm in the classical Galerkin method is added to establish a well-posed approximation problem. Moreover, due to the nonlocal Fisher-KPP structure, it is necessary to prove the positivity of the approximating solution in order to obtain uniform estimates. In the compactness arguments, the usual tightness argument in the framework of Hilbert spaces cannot be directly applied to the uniform estimates obtained in this paper. As a result, we develop a more general version of the compactness argument and tightness criterion, presented in the appendix, which will be applied throughout the paper. This allows for the global existence of nonnegative martingale solutions to be derived from Jakubowski's version of the Skorokhod Theorem, along with a thorough discussion of the convergence properties.

Optimistic Gradient Descent Ascent (OGDA) and Optimistic Multiplicative-Weights Update (OMWU) are two very popular algorithms to solve convex/concave saddle-point problems, where OMWU is the non-Euclidean, entropic version of OGDA. It is known since the '80s that the last iterate of OGDA asymptotically converges to a saddle point in smooth problems. On the other hand, it is unknown if OMWU has the same property. In this paper, I show that OMWU converges asymptotically for smooth convex-concave saddle-point problems, with a small enough constant learning rate. The result does not require uniqueness, strict complementarity, an error bound, or initialization near a solution. The main new ingredient is a boundary argument showing that every cluster point satisfies the inactive-coordinate KKT inequalities. The boundary argument was discovered with assistance from ChatGPT and is documented in the appendix.

Originally motivated by creating first-person computer visualizations within Riemannian manifolds – the author was led to study deformable-body mechanics, as rigid-body mechanics is not available in a generic Riemannian manifold due to its lack of nontrivial isometry group. Hyperelasticity is a particularly nice sub-category of continuum mechanics in which a deformable, elastic body's behavior is determined by a stored energy density function. This allows problems to be posed variationally, and powerful tools brought to bear on studying and solving them. This article presents numerical simulations of static solutions to a particular class of problems in hyperelastic mechanics in 2-dimensional Riemannian manifolds in which a flat hyperelastic body $B$ is embedded into a region $\Omega$ in a nowhere-flat surface $S$ of revolution $z=z\left(r\right)$ such that $\left|K\left(r\right)\right|$ decreases as $r\to\infty$, where $K$ denotes the Gaussian curvature of $S$. For example, the funnel $z=-r^{-1}$ or the paraboloid $z=\frac{1}{2}r^{2}$. Because $B$ is flat, the body can't achieve a zero-stored-energy configuration, and restorative forces arise in the body to move it toward a region of lower stored energy – meaning, toward a flatter configuration. With the addition of a gravitational potential $U\left(r\right)=z\left(r\right)$ on $S$, forces act on the body to pull it toward $r=0$. If the body has sufficient stiffness and remains within the region $\Omega$, then the body has an equilibrium configuration in which the body's deformation-response forces perfectly cancel the gravitational forces. Such a configuration represents a kind of "levitation" phenomenon within this surface. The numerical implementation of this problem will be detailed and the resulting numerical solutions and various consequences discussed.

In this paper, we propose a segment-wise soft robotic antenna (SRA) system, where each soft robotic arm referred to as a tentacle, comprises multiple independently controllable segments with bending, elongation-retraction, and sweeping motions. By adjusting segment motion parameters, the positions of surface-mounted antennas are reconfigured, distinguishing it from conventional reconfigurable antenna (RA) systems. Based on this model, we propose two antenna deployment schemes: the segmented end-antenna configuration (SEAC), where fixed antennas are mounted at the segment ends and reconfigured via segment motions; and the hybrid end-and-intermediate antenna configuration (HEIAC), where RAs are further integrated as intra-segment antennas. In HEIAC, soft-robot segment deformation provides large-scale spatial reconfiguration, while RAs enable fine-grained adjustment. For SEAC, we formulate a sum-rate maximization problem accounting for inter-segment connectivity and the nonlinear mapping from segment deformation parameters to antenna coordinates, and develop a penalty dual decomposition-projected gradient ascent (PDD-PGA) algorithm. For HEIAC, we jointly optimize segment deformation, intra-segment antenna positions, and antenna activation using a block coordinate descent (BCD)-PDD-PGA algorithm with greedy backward antenna selection. Simulation results demonstrate that the proposed schemes substantially outperform fixed-position antenna arrays and conventional RA baselines. In particular, SEAC and HEIAC achieve 37.9% and 32.1% sum-rate gains over conventional 3D reconfigurable arrays, respectively, while SEAC provides up to a 49.3% gain in compact array deployments.

Recently, constructions of linear complementary pairs (LCPs) of codes and linear complementary dual (LCD) codes on function fields have attracted considerable attention due to the wide range of applications of these codes. Such constructions rely on non-special divisors of degrees $g$ and $g-1$. In this work, we investigate Kummer extensions defined by $y^m = f(x)$ with $f(x)\in\mathbb{F}_q(x)$ and establish an arithmetic characterization of non-special divisors whose support can contain non-totally ramified places. Based on this characterization, we explicitly construct non-special divisors of degree $g-1$ on the GK curve. Moreover, utilizing pure gaps, we explicitly provide several families of effective non-special divisors of degree $g$ on Kummer extensions with the same multiplicities. We then develop a general framework for constructing LCPs of algebraic geometry (AG) codes on Kummer extensions. By virtue of canonical divisors, we show that the security parameters of LCPs of AG codes can be determined within this framework, which also enables the construction of LCD AG codes. Finally, we illustrate our results with representative examples, including LCPs of codes on the GK curve and LCD codes on quotients of the Hermitian curve.

The process of deriving the Desarguesian plane $PG(2,q^2)$ to get the Hall plane is well known, and the problem of when a conic in $PG(2,q^2)$ inherits to an arc in the Hall plane has been solved. In this article we look at the generalisation of replacing an André net of $PG(2,q^t)$, $t\geq 3$ to construct an André plane of order $q^t$. This article looks at the case where $q$ is odd and $t$ is prime, and determines when a conic in $PG(2,q^t)$ inherits to an arc in an André plane. Further, the number of arcs in an André plane that are inherited in this way is enumerated.

We prove exact Fourier-dimension formulas for scalar dyadic Mandelbrot cascades pushed forward to fixed nondegenerate $C^2$ embedded arcs and fixed nondegenerate $C^2$ Jordan curves in $\mathbb R^2$. Let $W$ be in the minimal Kahane–Peyriere regime. For each fixed nondegenerate $C^2$ embedded arc $\gamma:[0,1]\to\mathbb R^2$, the pushforward $\mu_\gamma$ of the interval cascade satisfies, almost surely on non-extinction, \[ \dim_{\mathrm F}(\mu_\gamma)=A_{\mathrm{loc}}(W), \] where \[ A_{\mathrm{loc}}(W) = \sup_{q>1} \max\left\{ 0,\, \frac{q-1-\log_2\mathbb E[W^q]}{q} \right\}, \] with the $q$-term interpreted as $0$ when $\mathbb E[W^q]=\infty$. The analogous formula holds for scalar circle cascades pushed forward by fixed nondegenerate $C^2$ Jordan curves $\gamma:\mathbb T\to\mathbb R^2$, with the pushforward denoted by $\mu_\gamma^{\mathbb T}$. This extends the scalar circle endpoint formula from the canonical circle to fixed parametrized arcs and Jordan curves. The main new issue beyond the canonical circle is the loss of the explicit trigonometric phase and, for arcs, the presence of endpoint stationary regimes. We prove the arc lower bound by a finite-$r$ annular Fourier theorem based on an endpoint-safe phase decomposition, phase-bin coefficient estimates, predictable capping, complex Freedman concentration, and an $r$-tail compensator. The Jordan lower bound follows by first-generation dyadic cutting into two fixed arcs. The matching upper bounds use deterministic curved-support obstructions together with the scalar-circle minimum lower local-dimension theorem.

Given a graph $G$, let $\chi(G)$ denote the chromatic number of $G$. For $k\in \mathbb{N}$, a graph $G$ is $k$-$vertex$-$critical$ if $\chi(G)=k$ and $\chi(G-v)< k$ for all $v\in V(G)$. A recent problem of Beaton and Cameron [TCS 1042 (2025) 115234] asks for which graphs $H$ of order five, are there finitely many $k$-vertex-critical (co-gem, $H$)-free graphs, for all $k\in \mathbb{N}$? Here we identify three distinct graphs on five vertices that yield an affirmative answer to this problem. More precisely, we show that for each $k\in \mathbb{N}$, there are finitely many $k$-vertex-critical (co-gem, $H$)-free graphs, where $H\in \{$paraglider, dart, house$\}$, by analysing the structure of such graphs. Our results together with a result of Couturier et al. [Algorithmica 71:1 (2015) 21–35] imply that for each $k\in \mathbb{N}$, there is a polynomial-time certifying algorithm for $k$-COLORING of (co-gem, $H$)-free graphs, where $H\in \{$paraglider, dart, house$\}$.

We study the full automorphism groups of smooth cubic fourfolds with prescribed symplectic automorphism group. Our starting point is the classification of symplectic automorphism groups by Laza and Zheng. We focus on the non-symplectic index, namely, the index of the symplectic automorphism group in the full automorphism group. We prove general restrictions on this index. We also compute bounds by group-theoretic and lattice-theoretic methods. In several cases, we determine all possible indices. For coinvariant lattices of rank 19, we classify all possible pairs consisting of the symplectic automorphism group and the full automorphism group.

In 'On Denoting' Russell proposed the most influential theory of definite descriptions, expressions of the form 'the F'. Characteristic for Russell's approach is that definite descriptions are not treated as what they appear to be on the surface, i.e. as singular terms. Instead they are eliminated by a contextual definition. Russell formalises definite descriptions in the context of complete sentences of the form 'The F is G'. This requires scope markers to distinguish, e.g., internal from external negation. It was recognised by Burge, and Kalish and Montague, however, that the essential features of Russell's approach may be formalised while respecting the syntactic category to which definite descriptions appear to belong. An alternative, favoured by Neale, follows Russell in that complete sentences 'The F is G' are formalised by a binary quantifier. The undeniable importance of the theory of definite descriptions for logic, mathematics and philosophy demands that it be formalised to meet the standards of modern proof theory. This is the topic of the present paper. We systematise, compare and extend existing approaches. After presenting its essential features, we formalise Russell's theory of definite descriptions in sequent calculus. Three approaches will be considered. The first uses a binary quantifier, whereas the remaining two employ the term-forming iota operator. The first of these employs only the iota operator, the other employs in addition the lambda operator which does duty as a scope marker. All systems satisfy the standards for modern proof theory, in particular cut elimination. The appendix reformulates these systems in natural deduction, which is more convenient for practical purposes.