Papers

We introduce sub-connectivity logic, denoted by $\textsf{FOscon}$, an extension of first-order logic for graphs with a specified set $Z$ of admissible edges. Its additional atom $\textsf{scon}(x,y;\bar z)$ asserts that $x$ and $y$ are connected by a path using only edges of $Z$ and avoiding the vertices in $\bar z$. Our main structural result states that, for every weakly sparse graph class $\mathscr C$, the class of all expansions of graphs in $\mathscr C$ by rooted spanning forest orders has bounded twin-width if and only if $\mathscr C$ excludes a fixed minor. When $\mathscr C$ excludes a fixed minor, we can also compute an additional linear order that preserves bounded twin-width in polynomial time. We translate $\textsf{FOscon}$ into first-order logic over an expansion by a depth-first spanning forest order of the admissible-edge graph. Combining this translation with our structural result, we obtain a model checking algorithm for $\textsf{FOscon}$ with running time $f(|\phi|,h(G))\cdot |G|^c$, where $f$ is computable, $c$ is an absolute constant, and $h(G)$ is the Hadwiger number of $G$. The translation and model-checking algorithm extend to binary relational structures, with the Hadwiger number measured on the Gaifman graph. For a graph class $\mathscr C$ let $\mathscr C_Z:=\{(G,Z):G\in\mathscr C,\ Z\subseteq E(G)\}$. We show that for every weakly sparse graph class $\mathscr C$, the class $\mathscr C_Z$ is monadically dependent for $\textsf{FOscon}$ if and only if $\mathscr C$ excludes a fixed minor. For monotone classes that admit efficient minor encodings, a corresponding hardness result makes this frontier computationally tight. We give applications to combinatorial reconfiguration and solution discovery problems, and study a guarded extension of $\textsf{FOscon}$ motivated by database queries.

We study peeling decoding on the binary erasure channel when redundant parity checks may be added without restriction in number, but each check is subject to a row-weight constraint. For random simple binary regular low-density parity-check (LDPC) codes, we consider the parity-check matrix formed by all nonzero dual codewords up to a prescribed weight. We show that, throughout an explicit logarithmic row-weight range, this complete family of redundant checks has the same asymptotic normalized-residual threshold as peeling with the original LDPC matrix. In the same range, with probability tending to one over the random code, it still has a linear-size stopping set that is uniquely decodable. Consequently, every parity-check representation whose peeling decoder succeeds on every uniquely decodable erasure pattern must contain checks above this logarithmic scale, and the supports of those heavier checks must collectively contain a linear number of coordinates. Thus row weight remains an asymptotic limitation even when the number and selection of redundant checks are unrestricted.

We construct a presentation of the Khovanov arc algebras of type $D$ by a quiver with quadratic relations. As an application, we use this presentation and the Koszul dual to compute some Hochschild cohomology groups of $\mathbb{D}_4$, and show that $\mathbb{D}_4$ is intrinsically formal, that is, it admits no non-trivial $A_\infty$-deformations.

We construct new classical global solutions of the one-phase Bernoulli free boundary problem in $\mathbb{R}^3$ whose positive phases are periodic perturbations of the product of the Hauswirth–Hélein–Pacard double hairpin domain with a line. In contrast with previous constructions of undulating cylinders, the free boundary of the base solution is not compact modulo translations. More generally, we show that the same scheme yields periodic solutions in $\mathbb{R}^{n+1}$, $n \geq 3$, bifurcating from product solutions $U(x,t)=u(x)$, when the base $u:\mathbb{R}^n\to [0,\infty)$ has nontrivial finite Morse index, is nondegenerate within a symmetry class, and satisfies certain natural geometric conditions at infinity. In dimensions $n\in\{3,4\}$, where the stable Bernstein theorem is available, the recent classification of finite Morse index solutions by Fernández-Real, Florit-Simon and Serra shows that every such solution satisfies these geometric conditions. Thus, the existence of undulating solutions in $\mathbb{R}^{4}$ and $\mathbb{R}^{5}$ bifurcating from their respective products reduces to a nondegeneracy property of the base solution. In order to confirm that the perturbed solutions are positive, we prove that finite Morse index solutions with noncompact free boundary satisfy a strong distance nondegeneracy estimate.

We show how to solve the conjugacy (search) problem and the problem of determining whether a given element is periodic in symmetric diagram groups based on finite semigroup presentations.

Let $X$ be an $n$-dimensional smooth complex projective variety admitting an int-amplified Galois endomorphism. We prove that its maximal rationally connected fibration is represented by a smooth toric fibration over a smooth $Q$-abelian variety. Consequently, if $X$ is rationally connected, then it is toric. If it also has Picard number one, then $X\cong \mathbf{P}^n$. As another application, we prove that such $X$ is of log Calabi-Yau type as conjectured by Gongyo.

In this paper, we introduce two types of two-sided slice regular functions, which are left slice regular in the first variable and right slice regular in the second variable. We prove that these two classes of functions coincide on axially symmetric domains. We further investigate fundamental properties of two-sided slice regular functions and define a star-product that preserves this regularity. In addition, we establish the Cauchy integral formula for these functions.

Fix a strip width \(\rho>0\), a Diophantine frequency \(\alpha\), an integer \(N\ge2\), and \(\lambda>1\). For the family \[ V_{\varepsilon,\delta}(x) =2\lambda\cos(2\pi N x)+\varepsilon\delta(x), \] we prove that there is an open prevalent set \(\mathcal P_{\rm strat}\) of real-analytic strip-\(\rho\) perturbations such that every \(\delta\in\mathcal P_{\rm strat}\) admits \(\varepsilon_0(\delta)>0\) for which \[ \begin{gathered} \operatorname{Leb}\left\{ E\in\Sigma(V_{\varepsilon,\delta},\alpha): \omega(\alpha,E;V_{\varepsilon,\delta})=s \right\}>0, 0<\varepsilon<\varepsilon_0(\delta), \qquad s=1,\ldots,N. \end{gathered} \] The proof relates acceleration to the total multiplicity of real zeros of analytic overlap coefficients for first-return cocycles. Under quantitative control of the return products, Poisson integration and Kac normalization identify acceleration with half this multiplicity. A positive analytic dual state gives the finite-volume spectral gaps and Schur curvature needed at the upper spectral edge. Schur variations separate the perturbed maxima, and parameter exclusion preserves \(s\) pairs of simple zeros on a positive-measure spectral set.

Identifying kinetic parameters and conserved quantities that support multistationarity and multistability remains challenging in chemical reaction network analysis. Structural theory provides existence and exclusion results, but does not always yield explicit, interpretable descriptions of the corresponding parameter regimes. We present a computational and machine-learning framework for discovering and characterizing parameter regimes with numerical evidence of multistationarity and multistability in mass-action systems. We refer to these numerically detected behaviors as approximate multistationarity and approximate multistability. The framework combines Latin hypercube sampling, adaptive sampling, and interpretable surrogate modeling. Logistic-regression and neural-network surrogates are trained separately on accumulated detection labels and independently guide later sampling toward regions with high scores for approximate multistationarity or multistability. Boundary refinement targets transitions between detected and undetected numerical classes. Buckingham–Pi groups are then used with CN2 rule induction to obtain interpretable empirical descriptions of detected regimes. Illustrative networks yield rules consistent with known multistationarity conditions, while the Schlögl model demonstrates adaptive enrichment of multistationarity detections. In the hybrid histidine kinase model and the sequential dual phosphorylation cycle, adaptive sampling substantially increased the number of parameter - total pairs with detected multistability relative to global sampling under equal sampling budgets. The framework complements analytical chemical reaction network theory by providing efficient, parameter-resolved discovery and characterization of multistationary and multistable regimes.

In this paper, we study the rigidity of Thurston's hyperbolic sphere packings on 3-dimensional manifolds. We prove that Thurston's hyperbolic sphere packing is locally determined by combinatorial scalar curvature. We further prove the infinitesimal rigidity that Thurston's hyperbolic sphere packing can not be deformed while keeping the combinatorial Ricci curvature fixed. The main tools include a characterization of the admissible space of Thurston's hyperbolic sphere packings, a variational formula for the dihedral angles of a tetrahedron and an identity for the variations of dihedral angles in a tetrahedron generated by Thurston's hyperbolic sphere packings.

A geometrically-exact planar beam finite-element framework is developed for the analysis of beams with hyperelastic constitutive models. A conventional three-field beam model is first formulated with full geometrical nonlinearity and a linear elastic material law. The kinematics are then enriched by an additional global field that permits deformation through the in-plane thickness. The out-of-plane direction is incorporated separately through plane-strain and plane-stress constitutive reductions, allowing the cross-sectional deformation assumptions and constitutive behaviour to be formulated independently. Numerical results for the present benchmarks show that the plane-stress condition gives essentially coincident centreline deformations for the Poisson's ratio values considered. In contrast, the plane-strain condition, which is widely employed in the literature, makes the formulation sensitive to Poisson's ratio coupling, with the beam becoming significantly stiffer at higher Poisson's ratios. Comparisons between Saint Venant-Kirchhoff and Neo-Hookean hyperelastic models further demonstrate close agreement in the global deformation and in-plane thickness response, with small constitutive differences becoming more apparent in the local out-of-plane stretch. The proposed approach offers a simple and consistent framework to integrate hyperelastic material models into planar geometrically-exact beam models.

For a quantum affine superalgebra with a fixed Drinfeld current presentation, we study shifted quantum affine superalgebras through a two-sided shifted Drinfeld current presentation, in which the leading degrees of the Cartan–Drinfeld currents are modified while the remaining current and Serre relations are retained. We then focus on the standard-parity type A algebras $\mathcal U_q^{\mu_+,\mu_-}(\widehat{\mathfrak{sl}}_{m|n})$. For arbitrary shifts and $m,n\ge1$, $m\ne n$, we prove a triangular decomposition. When $m,n>1$, we obtain PBW–Drinfeld bases for arbitrary shifts and a Levendorskii-type finite presentation of the one-sided algebra $\mathcal U_q^{0,\mu}$ for every antidominant $\mu$. For antidominant two-sided shifts, we construct shifted RTT realizations and prove the corresponding Ding–Frenkel isomorphisms. We construct a shifted formal Drinfeld coproduct for arbitrary two-sided shifts, coassociative in a completed tensor product. Shifted RTT realizations yield non-completed one-sided coproducts in the antidominant region, made explicit on finitely many generators by the finite presentation; injective shift homomorphisms extend them to arbitrary one-sided shifts. Their ternary coassociativity holds whenever the middle shift is antidominant, including boundary cases. Finally, for two-sided antidominant shifts $\mu_+,\mu_-\in\Lambda^-$, a shifted full general-linear Ding–Frenkel isomorphism identifies the special-linear simple-current algebra as the fixed algebra under common scalar rescalings and yields coassociative two-sided non-completed coproducts.

We consider the one-level density of zeros of the Riemann zeta function weighted by its special values on the line ${\rm Re}(s)=\sigma$, where $\sigma$ is a fixed real number such that $1/2\le \sigma\le1$. We show that the density function for the Riemann zeta function changes if and only if $\sigma=1/2$.

We obtain presentations of Borel subgroups of certain Chevalley groups and Kac-Moody groups of simply laced type. More precisely, we show that Borel subgroups of simply laced Chevalley groups over a commutative, unital ring (satisfying a minor technical condition), are the amalgamated product of their rank 2 subgroups, generalising a result of Tits over fields (different from $\mathbb{F}_2$). We moreover show that Borel subgroups of Kac-Moody groups over local Bézout domains also have a presentation as an amalgamated product of their rank 2 subgroups as soon as this holds for their rank 3 subgroups. In particular we deduce that Borel subgroups of simply laced Kac-Moody groups over a local Bézout domain (with residue field different from $\mathbb{F}_2$) have such a presentation if and only if they are of 3-spherical type.

Harnesses are increasingly unlocking the potential of large language model (LLM) agents by making their actions more controllable. In automatic heuristic design (AHD) for combinatorial optimization (CO), LLMs iteratively revise solvers using execution and verification feedback. However, these revisions often break previously validated solver behavior, making refinement unstable and costly. We characterize this pattern as feasibility oscillation and document it across CO benchmarks, AHD methods, and backbone LLMs. Tracing this oscillation, our analysis reveals that feasibility degradation often follows large solver revisions, while substantial feedback redundancy makes it harder to identify which failures deserve attention. Motivated by these findings, we introduce DiSH, a harness that jointly controls revision direction and scope. It selects representative failure evidence to guide each update and progressively narrows the revision scope as feasible behavior accumulates. Across nine CO benchmarks, DiSH consistently improves QYI scores and reduces feasibility oscillation, while roughly halving feedback token consumption relative to prior methods that use full feedback. These results establish DiSH as a strong baseline for solver refinement under harness control.

We prove that the Calabi flow extends past any finite time on compact Kähler surfaces if its Sobolev constants remain uniformly bounded. Moreover, we show the long-time existence of Calabi flow in certain Kähler classes on compact Kähler surfaces when the initial Calabi energy is strictly below an explicit threshold.

Rota-Baxter Lie groups are the group-level counterparts of Rota-Baxter Lie algebras of weight $1$. In this paper, we first establish a correspondence between representations of Rota-Baxter Lie groups and representations of Rota-Baxter Lie algebras of weight $1$ via differentiation and integration. We then develop a cohomology theory for Rota-Baxter Lie groups with coefficients in a representation by constructing an explicit cochain complex in arbitrary degrees. We justify this theory by proving a van Est type theorem, which relates the cohomology of a Rota-Baxter Lie group to that of its Rota-Baxter Lie algebra of weight $1$. As an application, we prove that equivalence classes of central extensions of a Rota-Baxter Lie group are classified by the second cohomology group.

In Part I, we introduced a general framework for constructing logical gates in quantum CSS codes from cohomology invariants, focusing on cup products and their generalizations from topological to qLDPC codes. In particular, we produced a family of $C^{k-1}Z$ logical gates that we called copy-cup on $k$ copies of the same quantum code implemented by finite-depth circuits. Here, we vastly extend this framework by introducing twists incorporating code automorphisms, higher twisted cohomology operations, and folding operations in relative cohomology. The framework includes operations lifting coefficients from $\mathbb{Z}/2$ to $\mathbb{Z}/2^k$, enabling all diagonal logical gates such as single-qubit rotations $\operatorname{R}(2\pi/2^k)$. Composing these operations with maps that select cohomology classes yields addressable logical gates. Examples include addressable $T$ gates on three- and four-tori and gates in qLDPC codes beyond topological ones. We introduce cohomology operations, such as the higher twisted Pontryagin powers, that to our knowledge have not previously appeared in the mathematics literature. These operations arise from $E_\infty$-structures on cochains, with explicit constructions using cyclic resolutions and the surjection operad. We expect these results to be of independent interest in homotopy theory. Our formalism applies much more broadly to chain complexes carrying suitable operadic structures, and we expect this operadic perspective to uncover further connections between homotopy theory and fault-tolerant quantum gates.

We present a unified, transformation-free linear regression framework tailored for ordinal compositional data, such as distributions across educational levels or aggregate Likert-scale survey responses. Traditional log-ratio approaches often obscure interpretability, ignore the inherent ordering of categories, and struggle with boundary zero values. To overcome these limitations, we propose a deterministic model constrained to the space of column-stochastic transformation matrices, naturally preserving the fundamental geometry of the simplex. By adopting the weighted 1-Wasserstein distance as the loss function, our method directly embeds the ordinal nature of the data into the estimation process. We provide a computationally efficient and globally optimal solution explicitly formulated as a Linear Programming (LP) problem. The framework systematically addresses four distinct regression scenarios, introducing a novel ordinal tensor product to rigorously handle interactions across multiple compositional predictors. Furthermore, we equip the model with a suitable regularization strategy and novel diagnostic tools, including a Wasserstein-based coefficient of determination ($R^2_W$) and an Order Preservation Index (OPI). Extensive simulation studies and two empirical applications demonstrate the robust predictive performance and interpretability of the proposed methodology.

Let $M$ be a complete Kähler manifold of complex dimension $n\geq 1$ whose Kähler form is $d$-bounded in the sense of Gromov. We prove that its Bergman space, namely the Hilbert space of $L^2$ holomorphic $n$-forms, is infinite-dimensional. The proof combines Gromov's construction of auxiliary holomorphic line bundles, parameter-uniform $\db$ estimates, and $L^2$ methods with singular weights.

We study tangent flows of noncollapsed Ricci flow limit spaces in the sense of Fang-Li. We prove that the regular part of every negative time slice is geodesically convex: any minimizing geodesic that meets the regular part has its entire interior contained in it. We also identify each negative time slice with the metric completion of its regular Riemannian part, thereby identifying the negative-time tangent flow with its associated metric soliton.

We first define an extended Mabuchi functional for a class of generalized Kähler metrics of symplectic type on a toric Fano manifold $(M,J,\mathbb{T})$, and prove it is proper. Then by showing the monotonicity of functional along the normalized generalized Kähler-Ricci flow for any initial $\mathbb{T}$-invariant generalized Kähler metric with its symplectic form in $2\pi c_1(M,J)$, we prove that the flow converges to a Kähler-Ricci soliton. The result confirms a conjecture of Apostolov-Streets-Ustinovskiy.

In this paper, we investigate Bohr-type inequalities for certain integral transforms of some bounded holomorphic operator-valued functions defined on a shifted disc. More precisely, we consider functions $f\in H^{\infty}(\Omega_\gamma,\mathcal{B}(\mathcal{H}))$, where $\mathcal{B}(\mathcal{H})$ denotes the space of bounded linear operators on a complex Hilbert space $\mathcal{H}$, and study the Bohr phenomena associated with the series representations in \eqref{e1.2a} and \eqref{e2.1}. Moreover, we determine the sharp radii $R_\gamma>0$ for which the corresponding majorant series satisfy the required inequalities for all $|z|\leq R_\gamma$. One of our main results extends some recent result due to Kumar and Sahoo \cite{Kumar-Sahoo-Meditarr-2023} to a large extent, and the other one is new in the literature

The present article is a generalization of our paper \cite{GP21}, together with new applications. The original problem came up in connection with the classification of singularities in positive characteristic. Then it is important that certain invariants cannot increase if we deform a given singularity. In \cite{GP21} we proved this for the completed fiber dimension under deformations with section. In general the nearby fiber contains however several singularities and in this article we prove that also the sum of the completed fiber dimensions behaves semicontinuous. This is well known for analytic or algebraic families. However, for families of power series the problem is more difficult, since the the modules defining the invariants are quasi-finite but not finite over the base space. In fact, in general the usual fibre dimension is not semicontinuous. However, if we pass to the completed fibres in a family of rings or modules over arbitrary Noetherian rings, we can prove that their dimension is semicontinuous. The proof is different from that given in \cite{GP21} and, though more general, even easier. Finally we apply this to prove the semicontinuity of several singularity invariants, such as the Milnor number and the Tjurina number in families of hypersurfaces and complete intersections. We end with a problem concerning the Milnor number of an isolated complete intersection singularity.

Code switching moves an encoded qubit between two codes whose transversal gates together are universal, but the switching operation itself is usually not transversal and can spread a single fault into an uncorrectable error. We study switching between two permutation-invariant codes on the same qubits, one with transversal Clifford gates and one with transversal gates from the non-Clifford binary icosahedral group. Both are compatible with transversal $X$, $Z$, and $F = HS^\dagger$, and the encoders with this property form a small linear space. We show that its codes of distance at least three are cut out by at most two quadratic equations, and that all of them have the same error geometry in the sense that their single-qubit error states have the same inner products. It follows that a switching unitary commuting with qubit permutations and with the shared transversal gates at every instant, and tolerating one single-qubit fault at an unknown time, exists if and only if the two codes are joined by a continuous path of distance-three codes. Such a switch can be made error transparent, so that a fault at any moment reaches the target code as the same single-qubit error. For real encoders, the distance-three codes are the solutions of a single quadratic equation, and on thirteen qubits, they form an explicit circle through two Clifford codes and an icosahedral code. We construct the switching unitary along this circle as a collective operator, leaving its synthesis into physical control open, and verify that its recovery infidelity per fault is at machine precision, whereas two natural alternatives leave average infidelities of $0.13$ and $0.85$. A survey over odd qubit numbers up to $29$ lists where such switches exist.

Surrogate modelling is important to enhance the computational efficiency of uncertainty quantification for physical processes involving random parameters. The advantage of Fourier neural operator (FNO) compared to conventional surrogates is its cross-resolution prediction capability. In the current study, a cross-resolution surrogate based on FNO is built for two-phase transient Darcy flows in porous media with random parameters. A multiresolution Sobolev learning method is proposed for FNO to enhance the cross-resolution prediction accuracy by incorporating first and second-order gradients in the loss function using training samples of different resolutions. The gradients are approximated by resolution-related finite difference schemes compatible with the grid for numerical simulation. Experimental results demonstrate the effectiveness of the new method in training FNO for building cross-resolution surrogates.

We prove that no single rank ordered adversary strategy is globally optimal for the prediction with expert advice problem with six or more experts, in both the geometric stopping and finite time horizon settings. The proof is based on establishing a leading order correction when one expert moves far ahead of the others. This allows us to connect optimal strategies between $n$ and $j<n$ experts and utilize recent results on the exact optimality set for the five expert problem.

Consider the nearest-neighbor antiferromagnetic Ising model on $\mathbb{Z}^d$, $d\ge2$, at inverse temperature $\beta$ and in a uniform external field $h$. Let $\beta_c(d)$ be the critical inverse temperature of the zero-field Ising model. We prove that for every $\beta>\beta_c(d)$, the Gibbs state is not unique whenever $|h|$ is sufficiently small. This extends a previous result of Dobrushin (1968) from sufficiently large $\beta$ to every $\beta>\beta_c(d)$.

In this paper, we study the existence of ground states for two-component attractive Bose–Einstein condensates with displaced harmonic trapping potentials \[ V_1(x)=|x-x_1|^2 \ \text{and}\ V_2(x)=|x-x_2|^2, \] where $x_1\neq x_2\in\mathbb R^2$. For intraspecies interactions $a_1,a_2\in (0,a^*)$, we focus on the critical interspecies coupling \[ \beta=\beta^*:=a^*+\sqrt{(a^*-a_1)(a^*-a_2)} \] and prove that there exists a separation threshold $d_c\in(0,\infty)$ such that the corresponding constrained minimization problem admits no minimizer when $0<|x_1-x_2|<d_c$, whereas it admits a minimizer when $|x_1-x_2|\geq d_c$. The threshold is given by $d_c=\Lambda_*^{-1/4}$, where $\Lambda_*$ is characterized by an auxiliary variational problem and attained. Our results extend the previous work of Guo et al. [J. Funct. Anal. 276 (2019)] and complete the existence classification of ground states for two-component attractive Bose–Einstein condensates with harmonic trapping potentials.

We prove Schwarz lemmas for \(\bar\partial_b\)-harmonic maps from complete pseudo-Hermitian manifolds into K"ahler manifolds of negative curvature. For maps of bounded generalized dilatation we bound the full and the horizontal differential. For CR maps a coupled Bochner identity yields rank-dependent horizontal estimates and a bound for the Reeb differential. On Sasakian sources and complex hyperbolic targets we classify the maps attaining the horizontal bound, bound the deviation from equality in integral form by the deficit at one point, and prove compactness of almost extremal sequences. As an application, a Siu-type divergence identity yields foliation, pluriharmonicity and rank rigidity under volume growth conditions.