We argue that dependent versions of McDiarmid's inequality are a useful but underutilized tool in mathematical statistics, learning theory and theoretical computer science. To make this point, we first highlight that approximate tensorization of entropy (ATE) implies McDiarmid's via the Entropy Method. Second, we derive McDiarmid's inequality for non-isotropic Gaussian random vectors $X \sim \mathcal N(μ, Σ)$ through ATE with a constant of the order of the condition number of $Σ$. We both independently obtain this ATE through a simple application of stochastic localization and also discuss how a more general ATE for the Gibbs sampler due to Ascolani et al., 2026 generalizes McDiarmid's-like concentration to strongly log-concave and log-smooth probability measures. We then apply the resulting concentration inequalities to resolve a question on the concentration of $\operatorname{sign}(X)$ posed by Simone Bombari, investigate Erdős-Rényi graphs under dependence and prove a Dvoretzky-Kiefer-Wolfowitz-type inequality for observations from a joint measure fulfilling ATE and continuous marginal CDFs. For the class of strongly log-concave and log-smooth measures, this result improves upon a prior Dvoretzky-Kiefer-Wolfowitz-type inequality for non-i.i.d. observations due to Bobkov and Götze, 2010, by establishing the expected $1/\sqrt{n}$-rate of convergence under weak dependence instead of $n^{-1/3}$.

In this note, we exhibit a group which has a right-angled Artin group as a finite index subgroup, but no finite index normal subgroup isomorphic to any right-angled Artin group. This answers a recent question of Vankov.

The Milnor map is the homomorphism from the unitary bordism ring to the unoriented bordism ring, halving the dimension, that maps the unitary bordism classes of the complex Milnor hypersurfaces to the unoriented bordism classes of their real points. In this work, we propose to generalize this construction to the equivariant setup and we show the existence of such a map for the equivariant unitary groups of the circle and the cyclic group of order two. Furthermore, we relate the kernel of these Milnor maps to the magnetic unitary equivariant bordism groups of free conjugations.

In this paper we consider the Dirichlet boundary value problem driven by the weighted Laplacian equation and considered on the Sierpiński gasket. Under suitable growth conditions imposed on the nonlinear term, we provide existence and multiplicity results employing the direct method, mountain pass theorem and some related multiplicity result.

We study the communication complexity of convex decentralized optimization over time-varying networks, where $n$ nodes hold private functions and must agree on the global minimizer using only synchronous exchanges with neighbors. The cost is the number of communication rounds to reach accuracy $\varepsilon$ -- a measure akin to round complexity in the LOCAL model, but constrained by nodes sharing only oracle responses. We prove a new lower bound of $Ω\!\left(χ_{\mathcal G} \sqrt{κ_g}\,\log\frac{n}{χ_{\mathcal G}}\log\frac1\varepsilon\right)$ communication rounds, where $χ_{\mathcal G}$ is the condition number of the network Laplacians and $κ_g$ that of the global objective, showing the round complexity attainable under uniform regularity cannot be matched in the nonuniform regime. The construction rests on spectral graph theory: we embed time-rotating star gadgets into the edges of an expander and patch them to preserve spectral connectivity.

We investigate the pseudoconvexity of level sets of the distance function. First, we show the formula of \cite{Li17} for the Diederich--Fornæss index can be extended to characterize pseudoconvexity. This extension implies that the rate of change of the Levi form is nonincreasing at weakly pseudoconvex points. Additionally, we demonstrate that perturbing the distance function by a factor of $e^{-t|z|^2}$ ensures all level sets sufficiently close to the boundary are strongly pseudoconvex for all $t>0$.

In this paper, we extend the recent Ionescu--Wainger multiplier theorem for the set of canonical fractions by Kosz, Mirek, Peluse, Wan, and Wright in several directions. First, we prove its weighted version, which allows us to combine a multifrequency setting with appropriate arithmetic weights. Second, we establish useful seminorm variants of the theorem. Third, we improve the norm upper bounds and, surprisingly, show that these bounds cannot be uniform in the size of the family of canonical fractions. Finally, we demonstrate how these refinements (especially handling arithmetic weights) can be applied by giving a short proof of Bourgain's pointwise ergodic theorem for polynomial iterates.

We prove that the only CROSSes that admit a (not necessarily homogeneous) almost quaternionic structure are ${\mathbb{H}} {\mathbb{P}}^n$ and ${\mathbb{C}} {\mathbb{P}}^2$.

We introduce a general theoretical approach to hierarchical spline spaces that replaces the classical constructive definition - based on basis selection - with a descriptive formulation in terms of regularity constraints. Specifically, we define generalized hierarchical spline spaces on multi-level domains as collections of piecewise functions satisfying hierarchical contact conditions across interfaces between refinement levels. The proposed framework applies to a broad class of local function spaces and relies on a minimal abstract requirement, the extension assumption, rather than on specific polynomial properties. Within this framework, we identify rules under which the hierarchical selection mechanism yields a complete basis, in the sense that it spans exactly the space characterized by the contact conditions. As an application, we consider Tchebycheffian spline spaces. We show that spaces generated by extended complete Tchebycheff (ECT) systems fit in this framework, thereby establishing the completeness of hierarchical Tchebycheffian splines. This demonstrates that the proposed theory naturally extends beyond the polynomial setting and provides a unified foundation for hierarchical constructions in more general spline spaces.

We give two combinatorial presentations for the algebraic cobordism ring $Ω^*(M)$ of the toric variety of the Bergman fan of any loopless matroid $M$. As a consequence of our presentations, we obtain an $Ω^*(\mathrm{pt})$-algebra isomorphism $Ω^*(M) \simeq CH^*(M) \otimes_{\mathbb{Z}} Ω^*(\mathrm{pt})$, where $CH^*(M)$ is the Chow ring of $M$ and $Ω^*(\mathrm{pt})$ is the algebraic cobordism ring of the point. This isomorphism generalizes, in part, the exceptional integral isomorphism between the Chow ring and $K$-ring of a matroid, studied in the recent works of Berget--Eur--Spink--Tseng and Larson--Li--Payne--Proudfoot. For a complex hyperplane arrangement $\mathcal{H}$, we prove that the algebraic cobordism ring of the wonderful variety $W_\mathcal{H}$ of $\mathcal{H}$ and the algebraic cobordism ring of the toric variety of the matroid underlying $\mathcal{H}$ are isomorphic, and that both rings coincide with the complex cobordism ring of $W_\mathcal{H}$.

In this paper, we continue our study, started in~\cite{CDS1}, of a two-dimensional variational model for ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes~$\Q$-tensor for the liquid crystal component and a magnetisation vector field~$\M$, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between~$\Q$ and~$\M$. We analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~$\eps$ tends to zero. While in~\cite{CDS1} we showed that the (rescaled) energy density for the~$\Q$-component concentrates, to leading order, on a finite number of singular points, in this paper we prove the energy density for the~$\M$-component concentrates along a one-dimensional rectifiable set. Moreover, we prove that the curvature of the singular set for the $\M$-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e.~the singular set for the~$\Q$-component. Crucial to our arguments will be the energy estimates and compactness results proved in~\cite{CDS1}.

In this paper, we introduce generalized phase retrieval (briefly, g-phase retrieval) and generalized norm retrieval (briefly, g-norm retrieval). Then, we present some properties of these concepts in Hilbert spaces, with special emphasis on R^n. Finally, the stability of g-norm retrieval for R^n is proved.

Multigrid methods are among the most effective frameworks for solving large-scale sparse systems. However, achieving their hallmark linear scaling and rapid convergence crucially depends on an effective smoother algorithm, whose design is often highly problem-dependent. This paper develops a new approach, referred to as \textit{cascading smoothers} due to their operation as an ordered sequence of single-step block-diagonal smoothers. Each level in the cascade is optimised to maximally damp the output of prior steps via a Frobenius norm minimisation of the corresponding error propagators. In particular, we develop an additive (resp., multiplicative) formulation analogous to Jacobi (resp., Gauss-Seidel). Applied within a standard multigrid V-cycle, we show they are remarkably effective across a wide array of problems, including finite difference, finite element, and discontinuous Galerkin discretisations applied to Poisson, elliptic interface, and Stokes systems as well as multiphase variants. In every case, cascading smoothers closely match or significantly outperform their optimally-damped classical counterparts, yet require no parameter tuning apart from a few discrete solver choices. Additionally, the approach is highly parallelisable and robust to geometric and operator complexities such as unstructured meshes and high-contrast coefficients.

We study a Cahn-Hilliard equation with anisotropic traction flux arising as a reduced continuum model of mechanically biased cell interactions in digit-forming organoids. For a regularized problem with strictly positive bounded mobility, we introduce a mixed finite element discretization based on an implicit-explicit treatment of the chemical potential. We prove existence of discrete solutions, establish exact mass conservation and a discrete energy inequality, and show convergence of the fully discrete approximations to a weak solution of the regularized problem. Numerical experiments illustrate the resulting dynamics and show the transition from classical coarsening to traction-induced fingering and protrusive growth. The computational study is complemented by mass and energy diagnostics, an energy-balance residual, fingering-onset and protrusion-count diagnostics, and a manufactured-solution convergence study.

Let $A\geq0$ be self-adjoint on a Hilbert space $H$, let $T_{t}=e^{-tA}$, and let $P$ be an orthogonal projection. Relative to the decomposition $H=PH\oplus P^{\perp}H$, write \[ T_{t}=\begin{pmatrix}C_{t} & V^{*}_{t}\\ V_{t} & D_{t} \end{pmatrix}, \] where $C_{t}=PT_{t}P|_{PH}$, $V_{t}=P^{\perp}T_{t}P|_{PH}$, and $D_{t}=P^{\perp}T_{t}P^{\perp}|_{P^{\perp}H}$. The compressed family $\left(C_{t}\right)$ consists of positive contractions but need not form a semigroup. Its defect is given by \[ C_{s+t}-C_{s}C_{t}=V^{*}_{s}V_{t} \] while the complementary block satisfies \[ D_{s+t}-D_{s}D_{t}=V_{s}V^{*}_{t}. \] Thus the failure of $\left\{ C_{t}\right\} $ and $\left\{ D_{t}\right\} $ to be semigroups gives two Gram kernels associated with the same off-diagonal maps. We treat these covariance defects as positive definite operator-valued kernels and use their Kolmogorov spaces to recover the hidden dynamics they encode. We then study short-time rescalings of $E_{s,t}:=V^{*}_{s}V_{t}$. The tangent kernel \[ F\left(s,t\right):=\lim_{\varepsilon\downarrow0}a\left(\varepsilon\right)^{-1}E_{\varepsilon s,\varepsilon t} \] has its own Kolmogorov space, and the lower-right block dynamics induces a positive self-adjoint contraction semigroup on it. The representing vectors of $F$ then satisfy an additive cocycle identity for this semigroup. This gives an intrinsic restriction on the positive kernels that can arise as short-time compression covariance tangents.

This is an expository note. We take the dyadic Cauchy-kernel identity of Castillo-Costin-Costin, a global rational/factorial decomposition built on the polylogarithm, and follow it down several specializations. In each direction it returns to a classical landmark: the polylogarithm duplication formula and Hurwitz's Fourier-series formula; representations of the zeta function at the special argument pi and at rational arguments, in the neighborhood of Hurwitz's multiplication theorem; the Hasse-Sondow globally convergent series; and, through its discrete scale invariance, the extra zeros of the Dirichlet eta function together with the harmonic-sum asymptotics of Flajolet-Gourdon-Dumas, with Dirichlet L-values emerging as the amplitudes of a log-periodic oscillation. The aim is unification: to exhibit one compact identity as an organizing center from which these classical results may be read off. We claim no new theorems; where an identity may not previously have been displayed in exactly this form, we say so and explain why it is nonetheless a recombination of known ingredients.

We prove that the riso-stratifications of singularities of affine algebraic sets introduced by Bradley-Williams--Halupczok are embedding independent and can be computed étale locally. Building on this, we upgrade their procedure, originally formulated in non-archimedean and model-theoretic language, into a sharp canonical and functorial stratification process for schemes of finite type over fields of characteristic 0. Our work enables comparisons with classical approaches to singularities and makes these stratifications available without requiring familiarity with the original methods.

The universal commuting dilation constant $C_d$ is the smallest constant $α$ such that every $d$-tuple of contractions dilates to a commuting $d$-tuple of normal operators with norm at most $α$. The work of several authors shows that $1.5438 \lesssim C_2 \leq 2$, and it has been asked on a few accounts whether $C_2 < 2$. We provide a positive answer that, in fact, produces a near optimal upper bound of $C_2 \leq \frac{2}{\sqrtϕ}$ where $ϕ$ is the golden ratio. This tightens the gap on the universal commuting dilation constant to $1.5438 \lesssim C_2 \lesssim 1.5724$. We also tighten the known upper and lower bounds on $C_d$ for arbitrary $d$-tuples.

This work is concerned with the study of lower bounds for various expressions related to the arithmetic functions $φ(n)$, $ψ(n)$ and $σ(n)$. Several explicit estimates are established.

We introduce the categorical Hopf map as a categorical principal bundle over the two-dimensional sphere with fibre the categorical circle of Nora Ganter. We investigate its connection to the Hopf map. We present a factorisation of the categorical Hopf map through the Hopf map and the basic bundle gerbe over the three-dimensional sphere. We discuss three equivalent constructions for the basic bundle gerbe over the three-dimensional sphere. Finally, we conjecture that the categorical group String(3) is equivalent to the categorical group of symmetries of the categorical Hopf map.

Borel and symmetric Borel derivatives are generalized derivatives defined through local averages of difference quotients. Distributional point values, in the sense of Łojasiewicz and its symmetric variants, are a classical way of describing the local value of a distribution. This paper connects these two ideas. Writing $T_f$ for the regular distribution generated by $f$, we prove that finite first and second symmetric Borel derivatives give symmetric distributional point values of $T_f'$ and $T_f''$, respectively. For the first symmetric derivative, Borel smoothness is used as a sufficient condition to pass from the symmetric point value to the full Łojasiewicz point value. We also prove that the one-sided Borel derivatives determine the right and left distributional point values of $T_f'$, and that the ordinary Borel derivative gives the full point value when the two one-sided averages agree. Examples show why the second-order symmetric result cannot be strengthened automatically.

In fuzzy-set theory, the notion of convexity has often been formulated through scalar reductions, typically via scores and aggregation functions. Although useful, such reductions may obscure relevant order-theoretic information in the codomain, especially in complex set-valued settings. This article develops a lattice-theoretic framework for convexity on lattice-valued mappings over point-convex segment-generated abstract convexity spaces. The framework separates the segment structure of the domain from the lattice structure of the codomain, distinguishing intrinsic convexity, defined through the lattice meet, from relational and observable convexities induced by preorders or scalar maps. The article characterizes when scalar observables preserve or reconstruct intrinsic convexity, and which codomain operators preserve it through their pointwise extensions. It then specializes the framework to hesitant fuzzy sets endowed with the symmetric lattice, recovering classical fuzzy and interval-valued convexities as natural restrictions. The structural result shows that the symmetric order cannot be represented by any finite family of scalar observables. This obstruction appears even on two-point typical hesitant fuzzy elements. Consequently, symmetric hesitant convexity cannot, in general, be reconstructed by any finite family of scalar observables. Moreover, every finite family of monotone scalar observables admits a three-point hesitant profile that is scalar-convex for all selected descriptions but not symmetrically convex.

We prove that the unknotting number of the knot 11n102 is 2.

As quantum computing moves toward scientific computing applications, nonlinear differential equations remain a central challenge since quantum evolution is intrinsically linear. In this work, we introduce a reduced basis algorithm (RBA) for polynomial nonlinear ordinary differential equations (ODEs) and spatially discretized partial differential equations (PDEs). After time discretization, the method composes the resulting polynomial update map over $m$ timesteps, identifies the reduced monomial basis appearing in this composed map, and constructs a linear RBA operator whose action recovers the exact $m$-timestep nonlinear dynamics. Thus, at the level of the chosen discrete update rule, the method introduces no additional approximation error beyond the time discretization error. The qubit number requirement is governed by the size of the reduced monomial basis. For an $n$-dimensional polynomial ODE system of degree $p>1$, the lifted register requires at most $q_m^{\mathrm{ODE}} = O(nm\log p)$ qubits in the full basis scenario. For PDEs discretized on $N^D$ grid points, a locality-based construction requires at most $q_m^{\mathrm{PDE}} = O(D\log N + n m^{D+1}\log p)$ qubits. Hence, the dependence on the grid size remains logarithmic, while the nonlinear overhead is controlled by local reduced basis size. The main computational burden is moved from the quantum computer to a classical preprocessing step, where the reduced monomial basis and RBA operator are constructed for the chosen timestep window. Through numerical tests on the Lorenz system and the one-dimensional Burgers equation, we verify that the RBA reproduces the corresponding discrete time nonlinear dynamics exactly, while exposing the trade-off between timestep composition, reduced basis growth, and locality.

Quantum-enhanced displacement sensing with bosonic systems is typically formulated assuming that the oscillator is cooled close to its ground state before nonclassical probe preparation. We investigate whether such near-ground-state initialization is necessary, or whether sensitive probes can instead be generated directly from thermal states. We analyze hot quantum probes produced by squeezing, number-raising, and Schrödinger-cat-state generation applied to thermal inputs. We identify two distinct mechanisms by which thermal mixedness can remain compatible with enhanced displacement sensitivity. First, projecting a mixed probe onto a definite parity sector removes the usual thermal suppression of the displacement quantum Fisher information, which can then increase with initial thermal occupation. Second, coherent superpositions of opposite displacements can retain sensitivity through coherence between their displaced components, even when the underlying state is mixed. We use these two mechanisms to classify hot-state protocols according to whether their sensitivity comes from parity selection, coherence between displaced components, or both. Finally, we formulate an experimentally relevant optimization problem comparing initial cooling with direct hot-state preparation under realistic decoherence and show that complete cooling is not universally optimal. Our results establish hot-state engineering as a route to quantum-enhanced bosonic displacement sensing without mandatory ground-state initialization.

Projected quantum feature maps provide a strategy for using quantum processors as feature generators for classical machine-learning models. Building on counterdiabatic Ising-glass and one-dimensional Heisenberg PQFMs, we introduce a generalized two-qubit Hamiltonian-based PQFM that provides a unified way to encode classical features through local Pauli fields and pairwise two-qubit Pauli interactions. This construction allows distinct classical variables to be embedded along different Pauli axes of the same qubit, increasing the information density of shallow circuits while remaining compatible with hardware constraints. We develop and implement these methods in pqfmlib, a publicly available Python library for constructing, executing, and benchmarking Hamiltonian-based PQFMs.We then benchmark the generalized Hamiltonian PQFMs against reference PQFMs on four biomedical classification datasets under a nested cross-validation protocol with paired statistical tests. Quantum features are generated using both IBM quantum processors with up to 156 qubits and statevector simulations. Our results show that the generalized two-qubit Hamiltonian family provides the most consistent pattern of statistically supported gains over matched classical baselines, although the performance of all methods depends on the dataset, encoding strategy, measured observables, and hardware conditions. These findings support generalized Hamiltonian PQFMs as a promising route toward near-term quantum utility.

Full tomography of an unknown quantum evolution is resource-intensive and often unnecessary when the goal is only to predict selected properties. This motivates the study of classical shadow estimation of unitary channels (CSEU), a task in which one queries an unknown $d$-dimensional unitary $U$ and stores classical data that can later be used to predict expectation values $\mathrm{tr}[O \cdot UρU^\dagger]$ up to additive error $\varepsilon$ for arbitrary input states $ρ$ and observables $O$. We propose a parallel, non-adaptive CSEU protocol using $\mathcal{O}(d\varepsilon^{-1})$ queries when the input states or observables have constant rank. This achieves Heisenberg scaling with respect to $\varepsilon$ and is query-optimal, as we prove a matching $Ω(d\varepsilon^{-1})$ lower bound that remains valid even with stronger access to the unknown unitary. Our query-optimal CSEU protocol provides a versatile and powerful tool for quantum learning theory, pushing the performance limits of several fundamental learning tasks, including unitary channel tomography, Hamiltonian learning, boundary-regime quantum channel tomography, Pauli transfer matrix learning, inverse-free amplitude estimation, pure-state property estimation, and shallow-circuit learning. Remarkably, we show that optimal unitary channel tomography can be achieved using only parallel queries, closing the gap between the best achievable efficiency of parallel and sequential tomography protocols. Together, these applications establish our framework as a fundamental tool for learning properties of quantum processes, particularly for certain key tasks that require high precision.

For general max-k-XORSAT with $k \geq 3$, no polynomial-time algorithm can do substantially better than random guessing on worst-case instances unless $\mathsf{P} = \mathsf{NP}$: approximating beyond the random-assignment value of $1/2$ is $\mathsf{NP}$-hard. The picture changes when each variable appears in at most $D$ constraints. In that bounded-degree setting, polynomial-time algorithms can provably beat the random baseline by an additive amount of order $1/\sqrt{D}$. For Boolean instances, this scaling is known to be optimal: the matching hardness result is due to Trevisan, while the corresponding algorithmic guarantee was established by Barak et al. Whether the same holds over general finite fields, and what it implies for quantum algorithms, has not been established. We make this connection explicit and extend the hardness to max-E$k$-LINSAT$(q,r)$ with bounded degree $D$ and over arbitrary finite fields $\mathbb{F}_q$, proving that it is $\mathsf{NP}$-hard to exceed $r/q + \mathcal{O}_{q,r}(1/\sqrt{D})$. These results provide the complexity-theoretic benchmark for the bounded-degree instances targeted by decoded quantum interferometry (DQI), QAOA, and classical heuristics. Any quantum advantage on bounded-degree instances is therefore confined to the constant prefactor. We further show that in the context of DQI and on $(k,D)$-regular instances, this prefactor is sensitive to the nature of the decoder: DQI with classical decoders faces an information-theoretic $1/\sqrt{D \log D}$ barrier that prevents it from matching the hardness scaling, while DQI with quantum decoders is compatible with the $1/\sqrt{D}$ scaling -- identifying quantum decoding as the key ingredient for matching the complexity-theoretic scaling with DQI.

Non-isometric encoding arises in various important contexts in quantum error correction, most notably in the finite-energy, non-ideal codewords inevitable in experimental realizations of continuous-variable codes, and holographic quantum gravity. In this work, we present a general and systematic theory of non-isometric quantum error-correcting codes. In particular, we employ the approximate quantum error correction framework to quantitatively study the fundamental limitations imposed by non-isometric encodings on the accuracy of quantum error correction and implementation of logical operations. We apply our theory to analyze GKP and tiger codes under energy constraints, and discuss the implications to holography.

Measurements can be used to monitor the evolution of quantum systems and may lead to a universally quantized time statistics. It is known that the mean return time is quantized for strong and indirect monitoring through the winding number of the return amplitude in a one-dimensional space. Here we discuss that under multi-channel strong or indirect monitoring, where the latter is achieved through ancilla coupling, the mean return time of a quantum walk in the projected subspace is also quantized. This reflects a universal time quantization for a higher dimensional evolution.