Papers

In this expository article, which is partially based on the lecture notes of a short course delivered by the second appearing author, we introduce the subfield of the Calculus of Variations that is concerned with the study of vectorial variational problems for supremal functionals, with emphasis on the associated PDE systems arising as extremality conditions. The scalar theory has a rather short history, first arising in the work of G. Aronsson in the 1960s. The vectorial case is even more recent and first arose in work of the second appearing author in the early 2010s. The Calculus of Variations in $L^\infty$ is an all-important field for numerous applications, but it presents serious difficulties and requires new tools, different to those required for the classical case of integral functionals and Euler-Lagrange equations.

We prove that the volume of a compact connected Kähler manifold with holomorphic sectional curvature at least 2 is bounded above by the volume of the Fubini-Study metric of constant holomorphic sectional curvature 2 on the complex projective space of the same dimension. Moreover equality holds if and only if the manifold is biholomorphically isometric to complex projective space. This answers a question posed by Xiong and Yang. Our approach also yields a different proof of Zhang's sharp volume estimate and Liu's rigidity theorem for compact Kähler manifolds with positive Ricci curvature. In fact, our main result states that the same sharp volume estimate holds under a new curvature positivity condition (mean RC curvature positivity), which is implied by both positive Ricci curvature and positive holomorphic sectional curvature. The definition of this condition was inspired by the work of Yang. The proofs in this paper are due to ChatGPT 5.6 Sol Pro, and the paper is merely an exposition of its output. The proofs has been verified by the authors and they take full responsibility for any errors.

We prove a lower bound for the Hausdorff dimension of weighted totally irrational singular vectors in affine-spanning common-base integral self-similar sets satisfying the open set condition. For the middle-third Cantor square, the bound improves the previously known explicit lower bound.

We study how many observations are needed to determine the causal direction between two linearly related variables. Classical LiNGAM theory shows that independent non-Gaussian disturbances identify the direction, but does not quantify the difficulty when the causal effect is weak or the disturbances are nearly Gaussian. Let $\beta$ bound the absolute structural coefficient from below, let $\nu$ measure each standardized disturbance's distance from Gaussianity, and let the disturbance scales lie in $[\underline\sigma,\overline\sigma]$. We prove the sharp local minimax law \[ N_2^\star(\beta,\nu,\delta) \asymp \frac{\log(1/\delta)} {d_\beta^2+\beta^2\nu^2}, \qquad d_\beta= \left[\beta^2- \left(1-\frac{\underline\sigma^2}{\overline\sigma^2}\right)\right]_+. \] Previous theory established population identifiability or assumed a fixed separation between the two directions. By contrast, we establish the sharp sample complexity as a joint function of edge strength, distance from Gaussianity, and scale uncertainty, and characterize when identification comes from non-Gaussian dependence or from covariance alone. The proof was independently generated with GPT-5.6 Sol in Codex's Ultra mode during a two-hour session. The human author supplied the prompt and was responsible only forchecking the proof and revising and polishing the manuscript.

We present a conservative, structure-preserving, finite-volume scheme for ideal MHD that ensures pressure positivity, remains applicable across all Mach and Alfven regimes, and handles general nonlinear equations of state. The scheme splits the MHD system into three sub-systems according to characteristic wave scales: an advective part for hydrodynamic transport, a magnetic part for velocity-field coupling, and a pressure part for pressure-velocity coupling. Nonlinear advective terms are explicit, while the other two sub-systems are implicit, yielding a mild, velocity-based CFL condition that is supported by extensive numerical evidence. This makes the scheme suitable for gas-pressure or magnetic-pressure dominated regimes and the incompressible limit. The implicit discretisation gives a pressure equation that reduces to an elliptic form in the low-Mach limit for ideal gases and general thermodynamics. Pressure positivity is ensured via a local conservation-preserving modification of the pressure-internal-energy relation, avoiding a posteriori clipping while preserving conservation. The divergence-free constraint is enforced exactly via constrained transport. Second-order accuracy is achieved with an IMEX Runge-Kutta time integration, TVD reconstruction for explicit fluxes, and central discretisation for implicit terms. The scheme is validated against numerous benchmarks, including high- and low-Mach regimes, strongly magnetised flows, and standard MHD shock problems in 1D and 2D, demonstrating accuracy, stability, and excellent shock-capturing.

We introduce the geodesic quantum $f$-divergences $D_f^t$, $0\leq t\leq1$, obtained from the affine-invariant geodesic between the standard and maximal relative modular operators. They reduce to the classical $f$-divergence for commuting states. The logarithmic generator yields geodesic relative entropies joining the Umegaki and Belavkin-Staszewski entropies, while the power generators yield $(t,\alpha)$-Rényi divergences joining the Petz and geometric families. Our first main result is data processing of $D_f^t$ for every finite operator-convex generator $f$ and every $t\in[0,1]$. In particular, this gives DPI for the geodesic relative entropies and for the $(t,\alpha)$-Rényi divergences when $0<\alpha<1$ or $1<\alpha\leq2$. Our second main result identifies equality in the DPI: for invertible states, every equality-determining operator-convex generator has, at each nonmaximal parameter $0\leq t<1$, exactly the Petz sufficiency class; at $t=1$, this changes to the generally larger maximal, or BS, class, which also coincides with the equality class of the divergences associated with the quadratic generator for every $t$. Our third main result is the corresponding collapse of invertible geodesic quantum Markov chains: for each of the three ordered conditional-mutual-information constructions, the $t$-quantum Markov chains are precisely the quantum Markov chains for $0\leq t<1$, whereas at $t=1$ they are the invertible BS quantum Markov chains, a class that can be strictly larger. We also determine parameter-monotonicity regimes and the intersections with the $(\alpha,z)$ family. We prove strengthened data-processing and reconstruction estimates; we compare the three conditional orientations; we establish continuity bounds under positive lower-eigenvalue assumptions together with complementary discontinuity results; and we give a capacity-per-unit-cost interpretation.

Spatially varying coefficients are the primary source of circuit complexity in quantum simulation of linear advection-diffusion-reaction equations. This work presents a resource analysis of single-step quantum propagators constructed using sparse FABLE block encodings, Quantum Singular Value Transformation, and linear-combination-of-unitaries. We derive theoretical estimates for qubit count, gate complexity, and circuit depth in terms of the spatial discretization, sparsity, and polynomial degree, and compare these predictions with synthesized quantum circuits for one- and two-dimensional variable-coefficient problems. The resulting analysis quantifies the cost of encoding realistic transport operators and provides practical resource estimates for near-term implementations.

In this work, based on the discrete ordinate method, we propose a robust block preconditioning strategy for the 3D nonlinear steady-state radiation transport equation with heat diffusion term. The presence of the diffusive term of the temperature equation prevents its elimination into a single equation for the radiation intensity. To overcome this difficulty, all physical variables are assembled into a single monolithic linear system. The heat flux and temperature are treated as independent variables in a mixed $H(\mathrm{div})$-conforming finite element formulation. The equation for radiation intensity is discretised by a discontinuous Galerkin method with upwind flux, where a vectorial finite element space is used to couples the radiation intensity in different directions within each element. We then construct a Newton-Krylov iterative solver to solve the nonlinear equations, for which the core part is efficient preconditioning. To accelerate the convergence of Krylov's method, three block preconditioners are constructed, corresponding to different levels of approximation of the coupling between the temperature and radiation intensity. $P_{\mathrm{Schur}}$ retains the full coupling. $P_{\mathrm{Split}}$ drops the conductive contribution to the radiation block. $P_{\mathrm{BJ}}$ neglects the radiation-to-temperature coupling, retaining only the temperature-to-radiation coupling. Numerical experiments demonstrate the mesh independence and robustness of the proposed preconditioners.

Let $M$ be a closed smooth manifold of dimension $d\geq3$. Given a compact metrizable Choquet simplex $\mathscr S$ and a bounded nonnegative affine upper semicontinuous function $\mathfrak e$ on $\mathscr S$, we construct a $C^\infty$ diffeomorphism $h$ of $M$, isotopic to $\operatorname{id}_M$ and supported in an embedded $d$-dimensional solid torus $D^{d-1}\times S^1$, with an isolated minimal invariant Cantor set $K$. The invariant-measure simplex of $h|_K$ is affinely homeomorphic to $\mathscr S$ with entropy function $\mathfrak e$, whereas every ergodic $h$-invariant measure not supported on $K$ is a Dirac measure at a fixed point. Consequently, the set of measure-theoretic entropies of ergodic $h$-invariant probability measures and the topological entropy of $h$ are \[ \mathscr H_{\mathrm e}(h)=\{0\}\cup\mathfrak e(\operatorname{ex}\mathscr S), \qquad h_{\mathrm{top}}(h)=\max_{p\in\mathscr S}\mathfrak e(p). \] The map $h$ is $C^\infty$-approximable by zero-entropy diffeomorphisms isotopic to $\operatorname{id}_M$. Taking $\mathscr S$ to be a singleton yields counterexamples to Katok's intermediate-entropy conjecture on every such $M$. We also construct such counterexamples $h_j$ and numbers $c_j>0$ with $h_j\to\operatorname{id}_M$ in $C^\infty$, $c_j\to0$, and \[ \mathscr H_{\mathrm e}(h_j)=\{0,c_j\}, \qquad h_{\mathrm{top}}(h_j)=c_j. \] Hence the intermediate-entropy property is not $C^\infty$ open among diffeomorphisms isotopic to the identity.

In this short note, we show that some well-known filtrations of infinite dimensional groups of Fredholm operators associated to certain perturbation classes are, in fact, Fredholm $\Delta$-filtrations (see Definition 5 below.) For example, let $\E$ be a separable infinite dimensional real Hilbert space. The group $\GLK(\E)$ of all invertible operators on $\E$ which are compact perturbations of the identity is the structure group for Hilbert Fredholm manifolds and bundles modeled on $\E$ \cite{ElwTr, Ksch, Mkhr}. Using an orthonormal basis, there are canonical inclusions of general linear groups: $$\GL(1) \subset \cdots \subset \GL(n) \subset \GL(n+1) \subset \cdots \subset \GL(\infty) = \varinjlim \GL(n) \subset \GLK(\E).$$ We show this is a Fredholm $\Delta$-filtration of the Fredholm manifold $\GLK(\E)$ with dimension sequence $\Delta(n) = \dim(\GL(n)) = n^2$, which was not discussed in the classical Fredholm manifold literature because of the rigid constraint that the dimensions of a filtration increase only by one.

The anisotropic Calderón problem of determining a smooth Riemannian metric from boundary measurements, up to a boundary-fixing diffeomorphism, remains open in dimensions $n\ge3$~\cite{uhlmann2009electrical}. We establish unique identifiability results in two complementary regimes. In the first, the identity principle for quasianalytic functions propagates boundary information and yields unique identifiability on compact manifolds without a prescribed product structure, including a partial-data consequence; under a prescribed normal geometry, quasianalyticity is needed only in the distinguished direction. In the second, suitable symmetry or one-sided ordering assumptions lead to unique identifiability at $C^\infty$ regularity with full or restricted boundary access. Taken together, the results exhibit a tradeoff among regularity, geometric structure, and boundary access: quasianalyticity supplies continuation when no global product structure is prescribed, while symmetry or one-sided order replaces that continuation at $C^\infty$ regularity.

Let $S_c=P_{(0,c)}QP_{(0,c)}$ be the one-dimensional sinc-kernel concentration operator, let $N_a(c)=\#{n:\lambda_n(c)>a}$, and set $\bar L=\log((1-\delta)/\delta)$. We prove, uniformly for each fixed $A>0$, the tail-side quantile formula $N_\delta(c)=c+\pi^{-2}\bar L\log(4\pi^2c/\bar L)+O_A(\log c+\bar L)$ for $6\le\bar L\le A\log c$. It yields corresponding additive formulas for the lower half and full plunge, with main terms respectively $\pi^{-2}\bar L\log(4\pi^2c/\bar L)$ and twice this quantity. An exact one-tail-coordinate selection gives, for fixed $A>0$, $d\ge1$, and $q\in(1/2,1)$, the one-sided tensor-product bound $\Lambda_\delta(c;d)\ge\pi^{-2}d c^{d-1}\bar L\log(4\pi^2c/\bar L)-O_{A,d,q}(c^{d-1}(\log c+\bar L))$ for $L_{d,q}\le\bar L\le A\log c$, where $L_{d,q}=\log(q^{-(d-1)}(e^6+1)-1)$; the tensor content is nontrivial for $d\ge2$. The analytic input is a signed growing-parameter sine-kernel determinant asymptotic: uniformly for $0\le\omega\le A\log s$, $\log\det(I+(e^{2\omega}-1)K_s)=4\omega s/\pi+2\pi^{-2}\omega^2\log(4s)+2\log|G(1+i\omega/\pi)|^2+O_A((1+\omega)^4\log^2s/s)$, where $G$ is the Barnes $G$-function. We prove this negative-coupling counterpart of the Bothner–Deift–Its–Krasovsky theorem by direct IIKS steepest descent. We also retain the uniform head-side results and use a two-way determinant reduction to obtain the moving-depth lower-half bridge bound with constant $1/(32\pi^2)$; extending it to the deeper range uses Kulikov–Dam Larsen and may require a smaller constant. These counting formulas are additive. Their errors become uniformly relative when $\bar L$ tends uniformly to infinity; fixed thresholds are covered separately by Landau–Widom. A Lambert-$W_{-1}$ formula is recorded only for the continuous main term, not for individual eigenvalues.

In this paper, we investigate whether Harper's better than square-root cancellation phenomenon still appears in Piatetski-Shapiro sequences, and when one should expect only square-root size. If $c=c(X)$ satisfies $(c-1)\log X\to 0$ as $X\to\infty$, then for Steinhaus or Rademacher random multiplicative functions we have \[ \mathbb{E}\left|\sum_{n\in\mathcal{N}^{(c)}(X)}f(n)\right|=o\!\left(\sqrt{|\mathcal{N}^{(c)}(X)|}\right). \] For Lebesgue almost every fixed $1<c<\frac{8}{7}$, we prove the multiplicative energy satisfies \[ E^\times(\mathcal{N}^{(c)}(X))=2|\mathcal{N}^{(c)}(X)|^2-|\mathcal{N}^{(c)}(X)|+o\left(|\mathcal{N}^{(c)}(X)|^2\right). \] Together with a largest prime factor estimate, this allows us to apply the criterion of Soundararajan and Xu and obtain a standard complex Gaussian limit for the normalized Steinhaus sum. To the best of our knowledge, this gives the first asymptotic formula for the multiplicative energy of Piatetski-Shapiro sequences. For fixed exponent argument, our proof combines metric arguments with the Van der Corput's $B$-process and a double large sieve estimate with Robert-Sargos spacing theorem.

We study the geometry that spectral duality induces on the simplex of reset distributions for absorbed Markov processes with geometric resetting. In the Fisher–Rao metric, selected by Čencov's theorem, the separatrix $\Sigma$ of reset-neutral distributions is rigid: under the square-root embedding $\pi\mapsto2\sqrt\pi$ its interior is a totally geodesic subsphere of the positive octant, itself a lower-dimensional Fisher–Rao simplex. The reset response reduces to a finite structure: the response functionals span a subspace whose dimension $r$ equals the number of active orbits of the duality involution, and the local separatrix is its annihilator. At the vertices of the simplex we prove a sign theorem valid for every $r$, which recovers the two-zone phenomenon of the first paper of this series. The invariant $r$ also settles the question of global orientation: for $r=1$, under a scalar sign condition satisfied by the canonical realisation, the sign of the response is fixed by the side of $\Sigma$; for $r\ge2$ the orientation cannot stay constant, and we give the criterion under which no global sign law holds, with counterexamples due to T. Newton showing that it is met in the abstract class. The biased random walk with multi-site geometric resetting realises the whole picture explicitly.

Discussed are Bertrand systems on two-dimensional spaces of constant curvature, i.e. sphere and pseudosphere (Lobachevsky space). These models often involve applying quantization techniques, such as those based on the Laplace-Beltrami operator, to the classical models and analyzing the resulting quantum behavior.

The symmetric space $X_n={\rm SL}(n,\Rb)/{\rm SO}(n)$ can be interpreted as the Teichmüller space of marked, unit volume, flat $n$-dimensional tori. It comes with a unique (up to scale) ${\rm SL}(n,\Rb)$-invariant metric $d_{X_n}$. In 1939 Teichmüller gave a modular interpretation of $d_{X_2}$ (the hyperbolic metric) in terms of an extremal mapping problem for quasiconformal dilatation. Such a modular interpretation for $d_{X_n}$ for $n\geq 3$ has remained unaddressed: the natural candidates - minimal quasiconformal dilatation, Lipschitz constant, or total energy - do not work. In this paper we give such a modular interpretation, two in fact. We introduce the total expansion $\TE(f)\in [0,\infty]$ of a Lipschitz map $f:M\to N$ between Riemannian manifolds, a notion related to the notion of “$k$-dilatation” developed by Gromov, Guth and others. For volume-preserving Lipschitz maps $f:\Tc_0\to\Tc_1$ between $n$-dimensional, flat, unit-volume tori, we prove that $\TE(f)$ is minimized in the homotopy class of $f$ precisely by the affine maps in that class and takes on these the value $d_{X_n}$. We prove similar results for the Hilbert-Schmidt expansion $\HE(f)$, which is a simple integral over $M$ and has more of an $L^2$ flavor.

For any group $G$, the set of all nonempty subsets of $G$ forms an additively idempotent semiring under set-theoretic union and elementwise multiplication, called the power semiring of $G$ and denoted by $\mathcal{P}(G)$. We prove that for a finite group $G$, $\mathcal{P}(G)$ has no finite basis for its identities if and only if $|G| \geq 3$. This completes the classification of the power semirings of finite groups with respect to the finite basis property.

We study spanning $H$-subdivisions in dense graphs where the length of every subdivision path is prescribed in advance. This problem is motivated in part by a question of Pavez-Signé [Combin. Probab. Comput. 33 (2024), 121–128], who asked whether the subdivision paths in a spanning $H$-subdivision can be required to have similar lengths. Let $h\ge3$ be an integer and let $0<\beta\ll\alpha\ll1/h$. We prove that, for all sufficiently large $n$, every $n$-vertex graph $G$ with $\delta(G)\ge n/2+\lfloor h/3\rfloor$ has the following property. For every graph $H$ with $h$ edges and no isolated vertices, write $E(H)=\{e_1,\ldots,e_h\}$, and every choice of integers $\ell_1,\ldots,\ell_h\ge4$ satisfying $\sum_{i=1}^h\ell_i=n-|V(H)|+h$ and $\sum_{\ell_i<\alpha n}\ell_i\le\beta n$, the graph $G$ contains a spanning $H$-subdivision in which the $i$th edge of $H$ is replaced by a path of length exactly $\ell_i$. We also give a family of examples showing that a linear additive term in $h$ is necessary in general.

We consider the problem of graphically local metric embedding, i.e. embedding points from an arbitrary finite metric space into a target metric space while preserving, up to a small distortion, only a subset of the pairwise distances specified by a bounded degree graph $G$. We provide a general reduction showing that, in many cases, this is no easier than embedding the points while approximately preserving all pairwise distances. As an illustration of our general reduction, we show that there exists a Euclidean metric space $X$ on $n$ points along with a graph $G = (X,E)$ of maximum degree $3$ such that any embedding of $X$ into $\ell_2^m$ which only preserves distances specified by $E$ up to a relative error of $(1+\varepsilon)$ must satisfy $m = \Omega(\varepsilon^{-2}\log n)$. Our lower bound matches the upper bound on the dimension coming from the Johnson-Lindenstrauss lemma for approximately preserving all pairwise distances; previously, such a lower bound was known only for the class of noncontracting embeddings [Schechtman-Shraibman, Discrete & Computational Geometry, 2009]. Moreover, the condition that the maximum degree of the graph is $3$ is best possible: for graphs $G$ of maximum degree $2$ (or more generally, treewidth at most $2$), any metric space embeds $G$-isometrically into any two-dimensional normed space.

We construct a fixed finite family $\mathcal L$ of ordinary forbidden subgraphs with $p(\mathcal L)=3$ and a constant $c>0$ such that $$ex(n,\mathcal L)>t_3(n)+cn^{3/2}$$ at every sufficiently large order. Nevertheless, the complement of every sufficiently large $\mathcal L$-extremal graph has at most two connected components. In particular, no such extremal graph is a complete join of three graphs of positive order. This gives a negative answer to a natural existence-only question motivated by the Simonovits Product Conjecture, in which one asks only for one product extremizer at each sufficiently large order.

In this paper, we investigate large values of quadratic Dirichlet character sums. We prove new Omega results for both short and long quadratic character sums under the assumption of the Generalized Riemann Hypothesis (GRH), which improve the previous results.

Motivated by Fisher's and Hill's \cite{Fisher,Hill} ideas of ranking and pivotal quantities, we introduce a planar Poisson process (PPP) model in which the negative component \(\Pi_-\), acted upon by a choice operator C, supplies a set of past benchmarks that divide the future points of the positive component \(\Pi_+\) into ranked categories. The benchmarks act as separators in an ordered paintbox while simultaneously acquiring the dual role of past standards established by the choice operator. The exceptional homogeneity properties of the PPP offer an infinitude of possibilities which absorb many existing combinatorial structures and enrich the toolbox of Bayesian distribution-free inference. A particular benchmark generating mechanism considered here (the exponential race) amounts to the device of splitting into spacings of nonhomogeneous order statistics. On the methodological side, the paper aims to highlight the role of order as important characteristic of an exchangeable structure, complementary to the description in terms of the components size. Moreover, we advocate the viewpoint that the order induced by a latent strength parameter is intrinsically inherited from the distant-past temporal sampling order, hence the decoupled orders may coexist within the framework of population duality without disturbing size-biasedness of components and the full exchangeability within the sample. This implies that the indistinguishability of components in the nonlinear CRP, sometimes regarded as nonexchangeability, does not in fact destroy exchangeability and should be reconciled with the arrival ordering within the paradigm of ordered structures.

We study tau-functions of elliptic and $ q $-Painlevé equations through their correspondence with 5d $ \mathcal{N}=1 $ supersymmetric field theories. Defining tau-functions associated with 5d field theories in terms of partition functions on the $ \Omega $-background, we propose a bootstrap method that determines their bilinear relations from the effective prepotential and the perturbative contributions, together with the global symmetry of the theory. Applying the method to 5d $ \mathrm{SU}(2) $ gauge theories coupled to $ N_f \leq 8 $ fundamental hypermultiplets, we obtain the bilinear relations of the elliptic and $ q $-Painlevé equations. This method can be applied to general 5d supersymmetric field theories beyond $ \mathrm{SU}(2) $ gauge theories.

We present a reinforcement learning (RL) approach to the co-design of stabilizer sets of Quantum Error Correcting Codes (QECCs) and decoders. We show how to produce a generative model that produces Bivariate Bicycle (BB) codes based on the choice of decoder. Specifically, we fix a decoder architecture and use Proximal Policy Optimisation (PPO) to train an agent over BB codes to maximise decoder performance under a depolarising channel noise model.

We obtain a closed explicit formula for the spectral determinant of flat conical metrics on the bielliptic genus-two locus. The metrics are generated by holomorphic one-forms with two simple zeros. For a symmetric reference metric, an exact Klein-four spectral identity reduces the spectral determinant to scalar determinants on spheres and tori; the singular anomaly formula then yields the general case. The resulting formula involves an elementary binary sextic in the coefficients of the one-form and two explicit hypergeometric areas of four-cone metrics on a quotient sphere. We apply this formula to the separating, one-node nonseparating, and simultaneous two-node degenerations of the curve and obtain complete asymptotics of the spectral determinant in all cases. The one-node degeneration has two distinct metric limits, according to whether the limiting one-form is holomorphic or meromorphic. Comparison of the cylindrical cases with the Bismut–Bost asymptotics determines the corresponding constants explicitly; in the separating case it also evaluates the relative determinant appearing in the Müller–Müller formula. As a by-product, the separating asymptotics evaluate the multiplicative constants left undetermined in earlier general conical and variational determinant formulas.

Solovay reducibility is studied intensively as a tool to compare the approximability and the degree of randomness of left-c.e. reals. By definition, a real is left-c.e. if it has a left-c.e. approximation, that is, it is the limit of an effective nondecreasing sequence of rationals. If reals $\alpha$ and $\beta$ have left-c.e. approximations $a_0, a_1, \ldots$ and $b_0, b_1, \ldots$, respectively, such that the approximation ratios \[ \frac{\alpha-a_n}{\beta-b_n} \] are bounded from above by a constant, the real $\alpha$ is Solovay reducible to $\beta$. The latter is the case for any such $\alpha$ and $\beta$ and their left-c.e. approximations whenever $\beta$ is Martin-Löf random by the Kučera-Slaman Theorem [DOI:10.1137/S0097539799357441]. This result was substantially strengthened by Barmpalias and Lewis-Pye [DOI:10.1016/j.jcss.2017.06.002], who demonstrated that, under the given assumptions, the approximation ratios are not only bounded but actually converge to a limit, which does not depend on the considered left-c.e. approximations. There is a quest for a suitable extension of Solovay reducibility to the class of all reals. Promising candidates include S2a-reducibility on the set of computably approximable reals by Zheng and Rettinger [DOI:10.1007/978-3-540-27798-9_39] and monotone Solovay reducibility by Titov [DOI:10.1007/978-3-031-95908-0_33]. For the latter, Titov [DOI:10.1017/jsl.2025.10157] demonstrated that the theorems of Kučera and Slaman and of Barmpalias and Lewis-Pye extend to all reals. He conjectured further [DOI:10.1017/jsl.2025.10157, Conjecture 3.2] that similar extensions hold for S2a-reducibility in terms of its functional characterization by Kumabe, Miyabe, and Suzuki [DOI:10.3233/COM-230486]. In this work, we refute this conjecture by proving that the analogue of the Barmpalias-Lewis-Pye Limit Theorem does not hold for S2a-reducibility.

Let $H_1, H_2$ be locally quasiconvex, torsion-free subgroups of a hyperbolic group $G$. We prove $\mathrm{rank}(H_1 \cap H_2)$ can be bounded in terms of $\mathrm{rank}(H_1)$ and $\mathrm{rank}(H_2)$.

We construct a six-functor formalism for syntomic cohomology of p-adic formal schemes. This, in particular, generalizes Poincaré duality to general smooth morphisms.

We study reductions of the tangent bundle of $S^{\dim G}$ through the adjoint representation of a compact connected Lie group $G$. If $d=\dim G$ and $r=\operatorname{rank} G$, we show that the adjoint map $\operatorname{Ad}\colon G\longrightarrow \mathrm{SO}(d)$ is homotopic, as an ordinary map, to one with values in $\mathrm{SO}(d-r+1)$. It follows that every vector bundle over a sphere associated to a principal $G$-bundle via the adjoint representation admits $r-1$ linearly independent sections. Combined with Adams's theorem on vector fields on spheres, this gives the necessary condition $r\le \rho(d+1)$ for an adjoint reduction of $TS^d$. In particular, no such reduction exists for a non-trivial compact connected group, simple or not, of dimension $d>1$ with $d\not\equiv3\pmod4$. As an application, this excludes the adjoint $E_7$-reduction of $TS^{133}$ that is the exceptional branch in the structure-group argument of Bor, Hernández-Lamoneda, Jiménez-Desantiago and Montejano for Banach's isometric subspace problem.

A graph \(G\) is said to be determined by its spectrum if every graph cospectral with \(G\) is isomorphic to \(G\). A T-shape tree is defined as a tree containing exactly one vertex of maximum degree three. For any three positive integers \(\ell_1\),\(\ell_2\) and \(\ell_3\) with \( \ell_1\leq \ell_2\leq \ell_3\), we denote by \(T(\ell_1,\ell_2,\ell_3)\) the unique \(T\)-shape tree such that deleting its degree-three vertex \(v\) yields three disjoint paths \(P_{\ell_1}\), \(P_{\ell_2}\), and \(P_{\ell_3}\), i.e., \(T(\ell_1,\ell_2,\ell_3)-v = P_{\ell_1}\cup P_{\ell_2}\cup P_{\ell_3}\), where \(P_k\) stands for the path graph on \(k\) vertices. In this paper, we establish a complete spectral characterization for the complements of \(T\)-shape trees, settling a long-standing conjecture posed by Wang and Xu (2006). Specifically, we prove that the complement of \(T(\ell_1,\ell_2,\ell_3)\) is spectrally determined if and only if \((\ell_1,\ell_2,\ell_3) \notin \{(\ell,\ell,2\ell-2):\ell\ge2\}\). Moreover, all cospectral mates of the complement of the \(T\)-shape tree \(T(\ell,\ell,2\ell-2)\) are identified for every integer \(\ell\geq 2\).