Papers

We present a comprehensive framework for tangent vector fields and differential operators on the rotation group $\mathrm{SO}(3)$ using harmonic series expansions, which provides a mathematical foundation for their implementation in the crystallographic texture analysis software MTEX. The central idea is to employ the standard left- and right-invariant frames as global orthonormal frames of the tangent bundle, thereby avoiding the numerical instabilities of the classical tangent space basis derived from the Jacobian of the Euler angle parametrization. While the choice of frames is primarily motivated by their geometric and numerical properties, their full potential emerges in the harmonic setting. Representing tangent vector fields through harmonic expansions of their frame components, we derive explicit frequency domain formulas for the gradient, divergence, and curl, allowing these operators to be applied directly to the harmonic coefficients. Moreover, we show that these differential operators preserve harmonic band-limitedness. In addition, the left- and right-invariant representations of tangent vector fields can be transformed into one another directly in the frequency domain, with an increase in harmonic bandwidth of at most one degree.

These lecture notes form the first part of a master's-level course on advanced numerical linear algebra. Their aim is not only to present the classical algorithms, but to show why the subject has become considerably more central than it was a generation ago. Numerical linear algebra grew up alongside the numerical solution of partial differential equations, and for a long time that is where its large sparse systems came from. Ranking the nodes of a network, assimilating observations into a weather forecast, and fitting a model to a large noisy data set now lead to problems of the same kind: too large to factorise, structured, and accessible only through matrix-vector products. Strikingly few ideas are needed for all of them. Each chapter therefore develops a standard topic and then puts it to work outside its original setting. We treat norms, factorisations, conditioning and floating-point arithmetic; sparse matrices arising from finite differences, from graphs and from machine learning; stationary iterations and the smoothing property; the conjugate gradient and Lanczos methods, with spectral clustering and regularisation by early stopping; Arnoldi and GMRES, with PageRank and large least squares; and finally preconditioning, Schwarz domain decomposition and multigrid. We assume a first course in linear algebra. Every section closes with a summary of what should be retained and every chapter with exercises, several drawn from past examinations. Accompanying Python code reproduces the numerical illustrations.

Anabelian geometry as an attempt to describe geometry in terms of étale topological data has addressed so far mainly categories of varieties over a field. In this paper we work over a normal base scheme $S$ of finite type over a sub-$p$-adic field and show that families of hyperbolic curves over $S$ are anabelian among smooth $S$-schemes with respect to dominant morphisms.

The game domination numbers $\gamma_{\rm g}$ and $\gamma_{\rm g}^\prime$, and the game total domination numbers $\gamma_{{\rm{tg}}}$ and $\gamma_{{\rm{tg}}}^\prime$ are investigated on Sierpiński graphs $S_p^n$. For the domination game, the trivial bounds arising from the known domination number and the Grundy domination number of $S_p^n$ are significantly improved by proving that $(2p-3)p^{n-2}\leq \gamma_{\rm g}(S_p^n), \gamma_{\rm g}^\prime(S_p^n) \leq (2p-2)p^{n-2}$. For the total domination game the bounds $\gamma_{{\rm{tg}}}(S_p^n), \gamma_{{\rm{tg}}}^\prime(S_p^n) \geq (2p-2)p^{n-2}$ are established.

Quantifying nonclassicality in continuous-variable systems remains a fundamental problem in quantum information science. The Tsirelson probability, central to the Tsirelson precession protocol, is defined as the average probability that a precessing quadrature yields a positive outcome when measured at $K$ equally spaced times, with the classical bound given by $1/2 \pm 1/(2K).$ In this work, we adopt this probability as a symmetry-sensitive probe to assess the quantumness of superpositions of Gaussian states in harmonic oscillators. We derive analytical constraints on Tsirelson probability arising from rotational and parity symmetries, and show that parity symmetry establishes a universal relation between the maximal and minimal Tsirelson probabilities within parity-related state families. Furthermore, we prove that even-fold rotational circular states cannot exhibit Tsirelson violation due to their definite parity. These results reveal how geometric symmetries of Gaussian-state superpositions determine their nonclassical behavior and provide a symmetry-based framework for probing quantumness in continuous-variable systems.

We prove a Poincaré inequality, uniform in the particle number, for repulsive (one-component) Coulomb gases in dimensions $d\ge3$, with inverse temperature $\beta_N$ that is allowed to depend on $N$. We assume a weak-coupling condition expressed in terms of the effective interaction strength $\Theta_N$ seen by a single particle. This yields exponential relaxation for the associated overdamped Langevin dynamics. The main ingredient is a one-site Poincaré inequality uniform in the number and positions of the Coulomb poles, including coincident poles. A moving-pole estimate and Dobrushin-Wu tensorization yield the many-particle lower bound, while the center-of-mass coordinate gives the matching upper bound.

We prove an integral representation of the $\alpha$-$z$ Rényi divergences for normal states on arbitrary von Neumann algebras. Along each ray $z=c\alpha$, the divergence is a positive average of relative entropies evaluated in a canonical family of escort states. For orders $\alpha$ above one, we require the divergence to be finite, whereas for orders below one support inclusion suffices. An explicit boundary correction removes the lower-order support assumption. The proof uses a compatible weighted Haagerup $L^p$ scale and applies outside the data-processing region. For faithful density matrices, we express the derivative of the escort relative entropy with respect to the Rényi order in terms of a nonnegative fluctuation quantity. Differences of order-one derivatives of the divergence give quantum Fisher information and Wigner-Yanase skew information. For faithful bipartite density matrices, we also express optimized conditional $\alpha$-$z$ entropies at orders above one as averages of ordinary conditional entropies. Further applications concern coherent-state divergences in quantum field theory and the optimizing order for the strong converse exponent in hypothesis-testing.

In this work we derive local differential Harnack estimates for positive solutions of filtration equations on complete Riemannian manifolds with a lower Ricci curvature bound by using Nash–Moser iteration. Inspired by earlier work on parabolic frequency functions, we establish some new frequency monotonicity formulas for these equations. The results cover the porous-medium and fast-diffusion equations, as well as several non-power diffusion laws.

Enrico Fermi in 1932 used classical Gauss equation to derive the Coulomb density–density interaction from the longitudinal electromagnetic potential. In this work we extend the Fermi procedure to the transverse component of the vector potential. By using a fully quantum canonical transformation, we replace the transverse vector potential with a current–current and current–current–density interactions. The transformed Hamiltonian is, then, projected in the fermionic space providing a matter–only, totally incoherent and gauge respecting Hamiltonian. The removal of coherence will avoid the breakdown of perturbation theory predicted by Haag's theorem, and make possible to introduce, in the transformed space, the diagrammatic approach. After discussing the implications of the transverse interactions on the current theories based on the Fermi procedure, we conclude by showing how the transverse Hamiltonian provides the quantum origin of the electromagnetic longitudinal–transverse splitting of phonons and other elemental excitations.

We study ramification ideals of finite Galois extensions of henselian valued fields whose Galois groups are products of pure subgroups. We give a criterion showing that the ideals supplied by the factors need not exhaust all subgroup ideals, and construct $C_p^2$-counterexamples of rank one with each defect $1,p,p^2$. For ramification-adapted families, whose intersections generate each ramification group, every subgroup ideal is represented by a cyclic subgroup of one member; when the members are pure, this gives a description in terms of distance sets. Such adapted decompositions always exist for elementary abelian groups. We also characterize principality of ramification ideals through defectlessness of the pure adapted factors. Kuhlmann's example shows that these conditions do not imply defectlessness of the full extension.

We study the support property for stability conditions obtained by gluing along semiorthogonal decompositions. Our main tool is a new gluing construction for abelian categories along a bimodule: under natural exactness assumptions the glued category is abelian, and it recovers the heart obtained by gluing $t$-structures. This gives a concrete description of objects in the glued heart which we use to prove the support property given a phase gap for glued semistable objects. In particular, we prove the support property assuming Collins–Polishchuk phase bounds. As applications, we construct stability conditions on dg-comma categories, including examples related to augmented curves in the sense of Alexeev and Kuznetsov, and analyze the behavior of the construction under mutations of semiorthogonal decompositions.

The factorizations over $\mathbb{F}_p$ of two supersingular polynomials $h_p(x)$ and $g_p(x)$ for genus $2$ curves, discussed by Ibukiyama, Katsura and Oort in their 1986 paper, are investigated. These polynomials are congruent modulo $p$ to the Jacobi polynomials $P_n^{(\alpha,0)}(1-2x)$, for $\alpha = \pm 1/4, \pm 1/6$, respectively. The number of their linear factors (mod $p$) is determined in terms of class numbers of the imaginary quadratic fields $\mathbb{Q}(\sqrt{-dp})$, where $d \in \{1,2,3\}$. The proofs use a quadratic transformation relating these polynomials to the Legendre polynomials $P_n(x)$; previous results on linear and binomial quadratic factors of $P_{(p-e)/4}(x)$ and $P_{(p-\bar e)/3}(x)$ proved by Brillhart and Morton; and properties of the irreducible quadratic factors of the class equations $H_{-3p}(X)$ and $H_{-12p}(X)$ (mod $p$). The quadratic transformation and linear factor results for $h_p(x) \equiv P_n^{(\pm1/4,0)}(1-2x)$ were first discovered using artificial intelligence.

Astrophysical binaries formed by a stellar-mass compact object and a massive black hole are among the most promising sources for future space-based gravitational-wave detectors. These extreme mass-ratio inspiral (EMRI) systems are tracked coherently over more than $10^{5}$ orbits, encoding detailed information about strong-field gravity. Translating a detection into physical parameters requires models of comparable precision. One outstanding difficulty is the treatment of transient orbital resonances. In this work we address the problem at the level of the Koopman operator, where the waveform is promoted to an observable and the evolution over one radial cycle is a linear operator. Removing the fast timescale without a singularity at a crossing is then a matter of choosing a function of that operator. We demonstrate the approach on a Kerr geodesic driven through the 3:2 crossing by a modelled forcing term. There the coefficient of the standard near-identity transformation diverges, while the finite-window generator attains its analytic bound to better than one part in $10^{6}$. Averaging over twice the crossing duration recovers the resonant jump with its predicted magnitude and phase dependence.

Let $G$ be a complex reductive group and $C_m(G)_1$ the identity component of the space of $m$-tuples of commuting elements in $G$. We prove that for every $m\geq 2$, the integral singular cohomology of $C_m(G)_1$ has torsion precisely at those primes which divide the order of the Weyl group of $G$. We deduce the analogous statement for compact Lie groups, settling a conjecture of Kishimoto and Takeda, and prove the corresponding result for compact Lie algebras. We prove these results using Smith theory for the conjugation action by semisimple elements of prime power order.

Pairwise reciprocal matrices are fundamental to the Analytic Hierarchy Process (AHP),a decision-making model. While the Direct Least Squares (DLS) method provides an intuitive mechanism for deriving priority vectors without complex transformations, it is susceptible to solution non-uniqueness. Under high levels of inconsistency, such as severe cyclic contradictions, the DLS optimization landscape becomes non-convex, yielding multiple distinct global minima. Consequently, priority rankings become unstable and critically dependent on initial algorithmic guesses. Furthermore, established prioritization operators (POs), including normalization techniques, the Eigenvector method, Singular Value Decomposition, Cosine Maximization, and the Pseudo-Inverse Gram Matrix (the closed-form solution of Weighted Least Squares), frequently generate disparate outcomes. To overcome these structural deficiencies, this paper introduces the Anchored Regularized Direct Least Squares (ARDLS) optimization model as a harmonizing framework. ARDLS integrates uniquely determined established POs as theoretical anchors within a regularization penalty. This integration systematically breaks mathematical symmetries and tilts the optimization landscape to guarantee convergence upon a single, unique global minimum. By minimizing the root mean square variance (RMSV) of the initial baseline vectors, ARDLS effectively unifies these divergent solutions. Comprehensive numerical experiments validate that the framework successfully fine-tune the solution of established POs by reducing RMSV while ensuring strict mathematical uniqueness. The practical utility of the method is further demonstrated through a numerical case study resolving an innovation fund dilemma in FinTech project selection. The proposed ARDLS approach offers a robust alternative to classical AHP across a wide range of decision-making domains.

In a wide variety of fields, every non-scalar matrix $M$ is similar to a matrix all of whose entries are non-zero. We give extensions of this result, including some in the division ring context. We then apply these results to the question of which pairs of operators generate the algebra of all operators on finite- and infinite-dimensional spaces. In the case of complex, separable infinite-dimensional Hilbert space, this is intimately related to the unsolved Invariant Subspace Problem. Finally, linear independence of subsets of arbitrary TVS's and generators of arbitrary TA's are shown to be stable in the sense of the theorems presented.

We study inverse problems for the elliptic and parabolic obstacle problems from boundary measurements. For the classical elliptic obstacle problem with strictly superharmonic obstacle function, we show that the Dirichlet-to-Neumann map admits a one-sided linearization at every boundary datum lying strictly above the obstacle. The linearized map is the Dirichlet-to-Neumann map for a rough Dirichlet problem on the a priori unknown non-contact set. We show that these linearized Cauchy data uniquely determine the non-contact set up to set of Sobolev $2$-capacity zero and consequently determine the obstacle. Our result applies to the inverse problem for a parabolic obstacle problem where both the coefficient and the obstacle function are time-independent by reducing to the elliptic inverse problem.

Taking classical information geometry as its point of departure, this paper investigates, through gradient flows, how dually flat geometry extends beyond regular convexity, non-degeneracy, and smoothness. The regular theory is developed from the log-determinant potential on positive definite Gram matrices, establishing its Legendre dual, Fisher–Rao metric, Bregman divergence, and generalized Pythagorean theorem. We connect this framework to Craig–Sakamoto deformation, Wolfe duality, and, via Yoshizawa's embedding, Brockett–Bloch–Ratiu double-bracket flows, linking isospectral dynamics, Stiefel optimization, and component learning. The Bures–Wasserstein geometry provides a complementary gradient-flow structure. The singular theory emerges from boundary behavior: difference-of-convex deformations produce indefinite or degenerate Hessians while retaining pseudo-Hessian, dually flat, Legendre-self-dual structures. Newton flows exhibit finite-time collapse or Łojasiewicz-controlled convergence near non-Morse critical sets. Fisher-metric degeneracies on the Birkhoff polytope and elliptic-curve moduli are resolved by explicit blow-ups, yielding a birationally invariant exponential decay law. We further derive a closed-form Kirillov Jacobian and introduce cross curvature as a spectral diagnostic of local escape rates, including a new Box–Cox interpolation. Reproducible numerical experiments support the closed-form results. Rather than claiming a completed theory, the paper provides foundations for singular information geometry centered on degenerate pencils, indefinite dual flatness, blow-up geometry, and Łojasiewicz-type convergence.

We investigate the variational properties of Riemann's ”nondifferentiable” function $R$. We show that $R$ has finite $p$-variation for $p>4/3$ and infinite $p$-variation for $p<4/3$. As an application of our results, we provide a new look on an upper estimate of the Hausdorff dimension of the image of $R$, due to Eceizabarrena.

Let $f:M^n\looparrowright\Sph^{n+q}(1)$, $n\ge2$ and $q\ge1$, be a closed, connected, non-totally-geodesic minimal immersion with second fundamental form $h$, and put $S=|h|^2$ and $S_*=\max_M S$. If $p\in f(M)$ has multiplicity $m$ and $f^{-1}(p)=\{x_1,\ldots,x_m\}$, then \[ \Vol(M)\ge \left[m+\varepsilon_n\sum_{j=1}^m \left(\frac{S(x_j)}{S_*}\right)^2\right]\Vol(\Sph^n), \] where $[110n(n+2)^2]^{-1}<\varepsilon_n<[104n(n+2)^2]^{-1}$. If the immersion is linearly full, then \[ \frac{\Vol(M)}{\Vol(\Sph^n)} \ge \max\!\left\{1+\varepsilon_n, \frac{4(n+1)^n}{(n+3)^{n+2}}(n+q+1)\right\}. \] Moreover, for every hyperplane $H$ through the origin, each connected component of $M\setminus f^{-1}(H)$ has volume at least $4(n+1)^n(n+3)^{-n-2}\Vol(\Sph^n)$; consequently the number of components is at most $\frac{(n+3)^{n+2}}{4(n+1)^n}\frac{\Vol(M)}{\Vol(\Sph^n)}$.

Sequent-style tableaux are a refutation calculus in which each node of the refutation tree carries a finite block of formulae and the structural rules are absorbed into the data structure and the closure criterion. In their original, classical form they rest on an involutive De Morgan negation and on closure upon a complementary pair. We show that both may be dispensed with. Replacing unsigned formulae by signed ones, we obtain a block calculus $\mathbf{B}_{ti}$ for intuitionistic propositional logic in which the whole of intuitionism is carried by one rule, the rule decomposing $\mathsf{F}(A \to B)$, which deletes the $\mathsf{F}$-part of the context on passing to the child block. The rules so obtained are, up to the presentation, those of Fitting's signed tableaux; what is new is the block format, in which the structural rules are absorbed rather than admissible, and what follows from it. We identify the semantic reason for this rule and for the one other anomalous one: of the signed compounds of the language, exactly those governed by the implication fail to be locally decomposable, and the two failures are repaired, respectively, by retaining the principal formula and by purging the context. We prove that $\mathbf{B}_{ti}$ is the multiple-succedent sequent calculus $\mathbf{G}_{ti}$ read upside down, that $\mathbf{G}_{ti}$ admits the structural rules, and that $\mathbf{B}_{ti}$ is sound and complete for Kripke semantics, with the finite model property and a block-theoretic proof of the disjunction property.

We provide an adjustment of the Hardy–Littlewood–Sobolev inequality for p=1 to the anisotropic setting. Several examples of anisotropic Bourgain–Brezis inequalities are obtained as corollaries of the main theorem.

Companion to arXiv:2608.XXXXX, which reduces threshold cascades of coupled Ornstein-Uhlenbeck diffusions on graphs converging Benjamini-Schramm to a unimodular Galton-Watson tree, unconditionally in a dissipative regime, to a finite-type Galton-Watson process. Two problems are left open there; we formulate both precisely, supply the analytic framework, and prove partial theorems. First, front propagation. We show the generation-indexed front admits a genuine branching random walk comparison, prove the front depth grows ballistically with an explicit speed $c_*$ given by the Perron root of a tilted mean matrix (unconditional in the dissipative regime), and reduce the conjectured Bramson delay $c_* t - \frac{3}{2c_*}\log t$ to the uniform integrability of a derivative martingale, which we construct at the linearised level. The front central limit theorem and the Bramson correction are stated as conjectures with a proof strategy. Second, the sensitive regime, where the type is a continuous failure strength and the offspring law is a Cox mixture. We prove the mean offspring operator $K$ on $L^2$ of the strength variable is quasi-compact with a spectral gap, the mechanism being Hilbert-Schmidt smoothing of the Ornstein-Uhlenbeck kernel governing strength inheritance. This yields a general-state-space Kesten-Stigum theorem, extends the $n^{-3/2}$ total-progeny law to continuous types with an explicit constant, and gives a strength-resolved central limit theorem. The continuous-type CRT scaling limit is reduced to a multitype invariance principle on a Polish type space, conjectured with all hypotheses verified modulo one tightness estimate. Neither problem is fully closed; each is given a framework, first-order theorems, and a precisely delimited remaining step.

Binary segmentation is widely used for multiple change-point detection because it is fast, simple to describe, and simple to implement. Its validity rests on the requirement that, at each recursive stage, the procedure identifies one of the true change-points when several are present in the current interval. This holds for detecting changes in mean using the CUSUM statistic, but fails in some other settings, in particular in slope change detection for continuous piecewise-linear signals. We propose Look-Ahead Binary Segmentation (LABS), a modification in which the change-points returned by the two child recursions define a narrower interval on which the parent estimate is re-evaluated. LABS inherits the computational speed of standard binary segmentation, but achieves the near-optimal consistency rate of $O\{(n\log n)^{1/2}\}$ in the slope-change signal setting when the LABS model is chosen via either thresholding or a Schwarz-like information criterion. Simulations show that LABS is fast and achieves state-of-the-art performance.

Classical fusion of sigma-model defects is commonly approached by combining their worldsheet actions and eliminating the intermediate fields. This procedure, however, does not systematically identify the resulting target-space defect or its possible branches. We address this problem by formulating fusion as a reduction of the target-space defect data. Local descent of the intrinsic two-form and global descent of the connection determine the admissible branches. For symmetry-preserving bi-branes in the $SU(2)$ WZW model and Hopf-fibre T-duality defects, the prescription reproduces the known biconjugacy-class branches and the finite family of fibre-shift defects. The appearance of several branches provides a classical manifestation of non-invertible fusion. We further use the framework to construct TsT and T-dual biconjugacy defects, including their global consistency conditions. It therefore provides a systematic method for computing classical fusion and constructing new defects from known ones.

We prove that any $n$-dimensional normed space is at Banach–Mazur distance $O(n^{2/3})$ from $\ell_\infty^n$.

In this paper, we construct weak Lyapunov functionals for kinetic Fokker-Planck equations with weak external confinements. This problem has remained widely open in the existing literature. Apart from showing interesting structures from the models, deriving these Foster-Lyapunov weights allows us to use Harris type arguments to show existence of steady states for these equations, and convergence to equilibrium with an explicit stretched exponential rate.

The Levin–Wen model provides a lattice realization of topological orders associated with a given unitary fusion category. A longstanding open problem is to characterize Levin–Wen ground-state wave functions intrinsically, without assuming a priori categorical symmetry data or a Hamiltonian. We address this by proposing six axioms on a family of wave functions defined on lattices at multiple scales. These axioms allow us to reconstruct the underlying unitary fusion category and prove that the resulting wave functions map to nonzero Levin–Wen ground-state vectors of the emergent category.

Let $\alpha>-1/2$, $\alpha\ne0$, and let $$ \Delta_\alpha=-\frac{d^2}{dx^2}-\frac{2\alpha}{x}\frac{d}{dx} $$ be the Bessel operator on $\mathbb R_+=(0,\infty)$. We characterize boundedness and compactness of commutators of $R_\alpha=\frac{d}{dx}\Delta_\alpha^{-1/2}$ on the Andersen–Kerman two-weight setting. For $1<p<\infty$ and $\mu,\lambda\in A_{p,\alpha}$, put $ \nu=\left(\frac{\mu}{\lambda}\right)^{1/p}, \ d\rho_\alpha(x)=x^{2\alpha+1}\,dx. $ For real-valued symbols, $$ \|[b,R_\alpha]\|_{L^p(\mu\,dx)\to L^p(\lambda\,dx)} \simeq \|b\|_{\rm{BMO}_{\nu,\alpha}}, $$ where the Bloom oscillation is computed with respect to $d\rho_\alpha$. Moreover $$ [b,R_\alpha]:L^p(\mu\,dx)\to L^p(\lambda\,dx) \text{ is compact} \quad\Longleftrightarrow\quad b\in\rm{VMO}_{\nu,\alpha}. $$ The proof uses the exact conjugation $ U_w(x)=x^{p-2\alpha-1}w(x), \ [U_w]_{A_p(\rho_\alpha)}=[w]_{A_{p,\alpha}}, $ which transfers the problem to the quotient Bessel kernel on $(\mathbb R_+,|x-y|,\rho_\alpha)$. We isolate the required Calderón–Zygmund estimates and one-sided non-degeneracy in this quotient normalization. The measure $d\rho_\alpha$ is distinct from the usual Bessel measure $dm_\alpha=x^{2\alpha}\,dx$. It is the Bloom base measure selected by the Andersen–Kerman conjugation and is used throughout the two-weight theory below.

Let $A$ be a commutative ring and let $I$ be an ideal. An element $a\in A$ is a regular denominator for $I$ when multiplication by $a$ on $A/I$ is injective; we denote the set of all such elements by $S_I$. We study how the common regular denominators for two ideals $I$ and $J$ are related to $I\cap J$ and $I+J$. This yields a particularly simple description when $I$ and $J$ are comaximal. Over Noetherian rings, the same viewpoint classifies denominator-equivalence classes by finite nonempty antichains of prime ideals, gives bounds for the associated primes of an intersection and leads to a local length identity. Furthermore, we show that flat base change preserves regular denominators, faithful flatness reflects them, and finite locally free quotients have fiberwise regular loci defined by determinants satisfying a multiplicative Mayer-Vietoris formula. Our results provide alternatives to primary-decomposition computations in many concrete situations.