Papers

Rigo and Salimov (2015) proved that the number of binary \(2\)-binomial equivalence classes of words of length \(n\) is the \(n^{\text{th}}\) cake number. We give a geometric explanation of this identity by constructing an explicit arrangement of \(n\) planes in three-dimensional space whose chambers are naturally indexed by these equivalence classes. This arrangement is the three-dimensional member of an infinite family of hyperplane arrangements. In dimensions \(1\), \(2\), and \(3\), the corresponding quotients recover abelian equivalence, a natural intermediate equivalence between abelian and \(2\)-binomial equivalence, and binary \(2\)-binomial equivalence itself. In higher dimensions, the same family realises natural refinements of \(2\)-binomial equivalence. We also determine the sizes of the resulting classes. Each size is given by a coefficient of a suitable Gaussian binomial coefficient. This yields the full class-size distribution for binary \(2\)-binomial equivalence and stabilisation results for the number of classes of any fixed cardinality.

We study the mosaic Almost Mathieu operator, a quasiperiodic model that naturally admits a singular strip-Jacobi representation. By establishing a duality framework and extending the correspondence between the integrated density of states and the fibered rotation number to this setting, we obtain an effective reduction to $SL(2,\mathbb{R})$ cocycles. As a consequence, combining Aubry duality, reducibility theory, and the Moser–Pöschel argument, we prove that the spectrum is a Cantor set for all noncritical parameters.

Chebyshev points are distinguished in polynomial interpolation by the logarithmic growth of their Lebesgue constants. This paper asks a simple question: how much can Chebyshev points be perturbed before they cease to behave like Chebyshev points? We study perturbed Chebyshev–Lobatto nodes $x_j=\cos(j\pi/n+\varepsilon_j)$, with angular perturbations $|\varepsilon_j|\leq \sigma_n$. The study is motivated by numerical experiments showing a broad stable region when the mesh fraction $n\sigma_n$ is small and rapid amplification for larger perturbations; the observed transition region is consistent with the curve $n\sigma_n\asymp(\log n)^{-1}$. The main result is a deterministic worst-case stability estimate: if $n\sigma_n(\log n+1)$ is bounded by a sufficiently small constant, then the Lebesgue constant remains logarithmic. The proof uses the cosine parametrization and Bernstein's inequality for trigonometric polynomials, thereby exploiting the angular geometry of the Chebyshev–Lobatto grid rather than a Markov inequality in the physical variable. We also give a worst-case obstruction at the angular mesh scale, showing that perturbations of order $1/n$ cannot be allowed uniformly. Consequences are derived for analytic interpolation in Bernstein ellipses, for the absence of Runge-type divergence in the stable analytic regime, and for pseudospectral differentiation. Numerical experiments illustrate the transition in the Lebesgue constants, the shape of the associated Lebesgue functions, Runge-function interpolants, and finite-precision differentiation errors.

We propose, analyze and test computational methods for solving a continuous-time heterogenous agent model with Epstein-Zin utility. Such recursive utilities allow the model to disentangle between risk aversion and intertemporal substitution. Having discretized the Hamilton-Jacobi-Bellman (HJB) equation arising in the model, we propose a Howard-Newton algorithm for the late resolution preference case, and a Howard-Tarski-Kantorovich algorithm for the early resolution preference case. We prove the convergence of the iterative algorithms. We obtain as a consequence the existence of solutions to the discretized HJB equations. In the late resolution case, we supply a priori estimates between the unique solutions of the continuous and discretized HJB equations.

The electrification of chemical process heating is essential to industrial decarbonization and sustainable manufacturing of chemical products. Joint optimization of electrified chemical process heating units and electric power systems is needed to achieve decarbonized operation of both sectors. In this work, we introduce a joint optimization model that identifies the optimal unit commitment of power systems and optimal operation of electrified steam cracking microgrids for olefins production. A mixed-integer linear programming (MILP) model is developed to optimize the hourly operational plan of 26 ethylene plants and the main power system in Texas. We propose a two-stage solution method to solve the resulting large-scale MILP problem efficiently. In the first stage, we apply Benders decomposition with LP-relaxed subproblems to decouple microgrid operations from the main power system. In the second stage, we use the first-stage optimal solution as a warm-starting point for the MILP. This two-stage approach reduces the average solution time by 93.5% compared to direct solution of the MILP. Results show that the largest overall greenhouse gas (GHG) emission reduction for both power systems and microgrids is achieved when the electrification level of steam cracking units is at 30%. Above 30% electrification level, a higher electrification level leads to higher overall GHG emissions and steady increase in operating costs, particularly on the microgrid side. Increasing renewables contributions in the electric power system helps debottleneck the electrification efforts and facilitate holistic decarbonization. We also remark that the optimal operational plan of electrified steam cracking microgrids also exhibit strong spatiotemporal patterns.

We prove Cusick's conjecture on the binary sum-of-digits function. More precisely, for every integer \(t\ge 1\) we show that \[ c_t:=\lim_{N\to\infty}\frac{1}{N} \#\{0\le n<N:\ s_2(n+t)\ge s_2(n)\}>\frac{1}{2}, \] and in fact obtain the explicit bound \[ c_t\ge \frac{1}{2}+2^{-2s_2(t)-1}, \] where \(s_2(m)\) denotes the number of ones in the binary expansion of \(m\). The proof is based on an exact deconvolution which replaces the distribution of \(s_2(n+t)-s_2(n)\) by a finite stopped random-walk law. The required bias is then proved through first-exit medians for principal subsequence ideals.

We develop a high-order, locally conservative, invariant-domain preserving continuous finite element method for nonlinear scalar conservation laws. A first-order graph-viscosity scheme is built on the fine-$\mathbb P_1$ submesh induced by the high-order nodes, so that it remains compact and has positive lumped masses for every polynomial degree, while the high-order candidate can be any stabilized Galerkin method; here we use residual-based artificial viscosity and continuous interior penalty methods. The two updates conserve different discrete masses, and comparing each in its own conserved measure gives a nodal difference with zero sum, which a graph-Poisson problem represents as antisymmetric edge fluxes. For polynomial degrees with positive $\mathbb P_k$ lumped masses (e.g., $\mathbb P_1$, $\mathbb P_3$, $\mathbb P_5$ on triangles), convex limiting of these fluxes conserves the $\mathbb P_k$ mass exactly and in local flux form. For degrees with zero or negative lumped masses (e.g., $\mathbb P_2$, $\mathbb P_4$), limiting perturbs the mass only on the elements where it acts, and this perturbation is removed inside each such element without violating the bounds, exactly whenever the element has enough room within its bounds. Numerical experiments demonstrate high-order accuracy for smooth solutions, robustness near discontinuities, and agreement with the entropy solution on standard benchmarks.

A catalogue of the Lie Amaldi Classification, together with their structures, is given using CLAUDE. It is explained in the paper why this structural description is correct and what guidelines were given to CLAUDE that made this possible. This catalogue will not be published elsewhere.

We prove that the Kim–Milman flow map enjoys favorable stability properties with respect to variations in the target measure, provided that one of the target measures is sufficiently regular. Our results include stability in relative entropy, and more notably, Lipschitz stability in the $2$-Wasserstein distance up to a logarithmic factor. We complement our results with a general existence theorem for these maps for any target measure with finite second moment.

We study local stable and unstable sets for two-sided continuous bundle random dynamical systems. For an ergodic invariant measure with positive fiber measure-theoretic entropy and positive fiberwise maximal Lyapunov exponent, we show that the fiber entropy is determined by the action of the random maps on local unstable sets, and establish a lower bound for the Hausdorff dimension of local unstable sets in terms of the ratio of entropy to the maximal fiber Lyapunov exponent. If the upper box dimension of the phase space is finite, we obtain a weak form of Ruelle's inequality.

We introduce a boundary analogue of the Gauss–Bonnet–Chern mass for asymptotically flat half-manifolds with non-compact boundary. We prove that this mass is well defined and establish the corresponding positive mass theorems for graphical and conformally flat graphs. Also we provide a Penrose-type inequality for the mass $\mathfrak{m}_{a,B}(g)$.

The ring of quaternions is defined over the field of real numbers. This article extends that framework by defining quaternions over a class of commutative unital rings and generalising several of their properties. It also studies the properties of the unit group of quaternions over commutative rings with special attention to halidon rings.

In this paper, we consider two classes of hyperkähler manifolds: moduli spaces of central-Levi parabolic Higgs bundles on the punctured sphere and star-shaped Nakajima quiver varieties. We produce a map $\mathcal T$ from a given star-shaped quiver variety $\mathcal X$ to a central-Levi parabolic Higgs bundle moduli space $\mathcal M$. We verify that $\mathcal T$ preserves stability and we show that it is a homeomorphism onto the locus of Higgs bundles with trivial underlying holomorphic structure. We then prove our main theorem: that $\mathcal T$ identifies the natural holomorphic symplectic structures on the two spaces. This theorem generalizes work by Biswas, Florentino, Godinho, Mandini from the rank 2, full flag, strongly parabolic case to arbitrary rank, partial flag, and weakly parabolic cases – namely, those whose Higgs field residues project to the centers of their respective Levi subalgebras.

Institutional rebalancing is a batched optimization workload with a hard operating deadline: hundreds of accounts need new weights under budget, turnover, exposure, exclusion, and tax-aware controls before trading can proceed. This paper evaluates Asymmetry PRISM, a CPU/GPU portfolio optimization engine, through a public evaluation boundary; problem data in, and returned weights, status codes, timings, memory class, external feasibility diagnostics, eligible objective comparisons, and audit records out. Within that boundary, the evaluation protocol fixes hardware and software versions, declares timing lanes, separates cold single calls from repeated workloads, and admits objective-gap claims only where an eligible reference solver completed. On completed multi-solver rows from N=100 to N=2,000, Asymmetry PRISM-CPU is 4.5x to 24.1x faster than the fastest completed reference row in the same lane. In the production queue study, Asymmetry PRISM-GPU completes 500/500 accounts over a 10,000-instrument universe in 109.5 s within a declared 25-minute operating window, with zero missed deadlines and an audit record for every solve; the recorded OSQP queue baseline completes 4/500. On an operationally constrained real-data suite (tax-motivated transition penalties, restriction caps, turnover controls, batches), Asymmetry PRISM clears constrained solves 3.4x to 126.7x faster than the best completing incumbent at certified-equal objectives, and the GPU route widens to 8.8x over the CPU route at N=384,800. Rows without a completed reference are reported as feasibility, timing, memory, and failure-status evidence.

We study fixed-cardinality subset selection for the exact integral bi-objective $R_2$ indicator with a uniform continuum of weighted Tchebycheff scalarizing functions. The indicator measures the area under the lower envelope of scalarizing losses over weight space, rather than a finite sample average over weight vectors. For a sorted bi-objective Pareto-front approximation, represented by points ordered by increasing first objective and decreasing second objective, we derive an exact adjacent-neighbor decomposition of this integral objective into boundary terms, unary diagonal corrections, and selected-neighbor transition terms. This yields an exact Bellman dynamic program with $O(kn^2)$ running time for selecting $k$ of $n$ candidate points. We then prove that the transition matrix is Monge. This gives a divide-and-conquer implementation with $O(kn\log n)$ running time and, more strongly, a staircase matrix-search implementation with $O(kn)$ running time under constant-time arithmetic comparisons. The matrix-search proof is presented through a lower-envelope sweep over single-crossing transition functions and includes the triangular feasibility condition $i<j$. The algorithms are exact for the continuous integral $R_2$ setting and are distinct from finite-weight-vector approximations, although they are related to earlier exact and dynamic-programming work on two-dimensional indicator-based subset selection, including hypervolume subset selection. Reproducible Python code compares exhaustive enumeration, the direct left-to-right dynamic program, the divide-and-conquer dynamic program, and the matrix-search implementation under explicit consistency checks.

Training dynamics is central to understanding neural networks, yet its theoretical analysis remains difficult even for simple architectures and becomes substantially more challenging for general modern architectures. In this paper, we propose a convergence framework for analyzing gradient descent (GD) dynamics under a broad family of neural network architectures and datasets beyond the neural tangent kernel (NTK) regime. The framework is formulated at the level of network blocks and covers architectures including pre-normalized multi-layer transformers. More precisely, under mild assumptions, we prove that for almost all initializations, GD with regular learning rates converges to the neighbourhood of a stationary point. This is mainly proved by establishing an iterate-dependent PL-type inequality through analyticity and measure-zero arguments, and by proving Lipschitz smoothness along the GD trajectory through polynomial generalized smoothness and a local relaxed dissipative condition. We further interpret the theorem under Xavier initialization and practical architectural scaling, showing that the learning rate scale depends on the depth and effective bottleneck dimensions rather than the largest width. Finally, we derive structural nondegeneracy implications for residual connections and function composition, and provide a generic characterization of global minimizers within our framework.

We study the fractional porous medium equation on connected infinite graphs with no local finiteness assumption. We introduce a notion of weak dual solution adapted to the discrete setting, and establish existence results for nonnegative initial data belonging to a weighted space defined through the fractional Green function, extending beyond the classical $\ell^1$ framework. Our approach relies on weighted estimates and on a detailed analysis of the associated fractional Green function. In the particular case of infinite trees with standard weights, we establish comparison principles and derive estimates for the fractional Green function, which lead to quantitative smoothing effects for solutions.

Dynamic feedback linearization (DFL) is a classical technique for trajectory tracking of unicycle-type mobile robots, but the resulting DFL-based controller becomes singular when the linear velocity vanishes, rendering standard DFL-based controllers unsuitable for stop-and-reverse maneuvers. This paper proposes a quadratic-program (QP)-based optimal control framework that avoids this singularity, while establishing local Lipschitz continuity of the resulting feedback law. Our approach reformulates the DFL constraints as an equality-constrained QP with a slack variable, ensuring feasibility for all states and reference signals, including at points where the robot's velocity vanishes. By introducing slack variables and tunable parameters, we demonstrate that the singular configuration can be avoided for a large class of reference trajectories. The effectiveness of the proposed approach for trajectory tracking is demonstrated through ROS 2-Gazebo simulations on a TurtleBot3 Waffle robot. The code is available at https://gradslab.github.io/DFL_QP_Unicycle/

We study differential polynomial rings $R[t;\delta]$ over $\Gamma$-graded rings, where $\Gamma$ is an arbitrary group. We show that $R[t;\delta]$ admits a $\Gamma$-grading compatible with that of $R$ if and only if $\delta$ is a $\gamma$-derivation for some $\gamma\in C_\Gamma(\Gamma_R)$, and that this grading is unique once $\deg(t)=\gamma$ is fixed; if $\delta\neq0$, then $\gamma$ is itself uniquely determined by $\delta$. We characterize the resulting graded ring by a universal property. We prove a characteristic-free center criterion for gr-simplicity whenever $Z(R[t;\delta])$ is a graded subring; in characteristic zero, gr-simplicity is equivalent to $\delta$-gr-simplicity of $R$ and $\gamma$-outerness of $\delta$, extending Jordan's simplicity criterion to the graded setting. We further show that $R[t;\delta]$ is gr-prime if and only if $R$ is $\delta$-gr-prime, and that gr-Noetherianity of $R$ passes to $R[t;\delta]$, recovering a graded Hilbert basis theorem as a special case. When $\Gamma$ is abelian, gr-simplicity and gr-primality are shown to be invariants of homogeneous graded Morita equivalence, and every ring homogeneously graded equivalent to $R[t;\delta]$ via a compatible idempotent is again a graded differential polynomial ring.

We study the fixed-density semi-inducibility profiles of the red-blue star $S_{2,1}$, which has one distinguished center, two red edges and one blue edge. For an $n$-vertex graph $G$, let $N(S_{2,1},G)$ be the number of injective labeled copies in which the two red edges of $S_{2,1}$ are mapped to edges of $G$ and its blue edge is mapped to a non-edge of $G$, that is, \begin{align*} N(S_{2,1},G)= \sum_{v\in V(G)} d(v)(d(v)-1)(n-1-d(v)). \end{align*} For every fixed red edge density $\beta\in[0,1]$, we determine both extremal $S_{2,1}$-densities. On the upper side, we prove the missing low-density range and, together with the theorem of Balogh, Lidický, Mubayi, Pfender and Volec for $\beta\ge 1/4$, obtain the full four-branch profile predicted in their work. On the lower side, we show that the natural endpoint profile coming from the quasi-star and quasi-clique constructions is not universal; the correct minimum is given by a one-parameter three-class complement-split family. The proofs use a transfer argument with degree-square tie-breaking, reducing the extremal analysis to almost-regular, threshold and finite-staircase optimizations.

The Mori-Zwanzig formalism provides a systematic framework for deriving reduced-order model of dynamical systems when only part of the state is observed, but its practical use is often limited by the complexity of the resulting computations. This paper develops an explicit formulation of the Mori-Zwanzig equation for linear time-invariant systems under partially observed observables. By expressing the dynamics in terms of observables, the Koopman generator, and projections onto resolved and unresolved components, we derive closed-form representations of the Markovian, noise, and memory contributions that arise in the Mori-Zwanzig identity. For the linear setting, the resulting formulas recover the reduced dynamics obtained from the variation-of-constants formula while retaining the operator-based structure of the Mori-Zwanzig approach. This makes the derivation a transparent reference case for reduced-order modelling with memory and clarifies how unresolved variables influence the observed dynamics through history-dependent terms. The analysis also identifies the ingredients needed for extensions to nonlinear systems and more general projections, including spectral filtering and data-driven approximations of memory effects. Analytical and numerical examples involving the harmonic oscillator and wave equations illustrate the construction and demonstrate how the formalism can be used to obtain interpretable reduced-order models for partially observed systems.

We develop a graphical calculus for induction and restriction along the Temperley-Lieb tower at a generic parameter. The main object, a diagrammatic 2-category $\mathfrak{D}$, has one-step generators that model the usual induction and restriction bimodules and additional two-step generators that model the summands that the cup-cap idempotents cut out in the two-strand Temperley-Lieb algebra. We construct an incarnation 2-functor from $\mathfrak{D}$ to the 2-category of Temperley-Lieb bimodules and prove a basis theorem for all 2-morphism spaces; the basis elements are indexed by bridges, a class of paths generalizing Dyck paths. The basis theorem implies that the incarnation functor is locally full and faithful; after we pass to the additive Karoubi envelope, this functor becomes an equivalence onto the corresponding 2-category of bimodules. Thus the calculus gives a concrete diagrammatic model for the functorial representation theory of the Temperley-Lieb tower. We also compute the split Grothendieck ring of the associated monoidal category and its action on the Grothendieck group of Temperley-Lieb modules. Under this action, homogenized Chebyshev polynomials represent the classes of standard modules and yield a positive integral basis. In contrast with Heisenberg categorifications, idempotent completion leaves the Grothendieck ring unchanged.

In 1975, Hitchin proved that any compact complex surface admitting a Kähler metric with positive holomorphic sectional curvature $HSC>0$ is rational. Conversely, he constructed such metrics on all Hirzebruch surfaces $\mathbb{F}_k$, as a first step towards characterizing rational surfaces by the existence of a Kähler metric with suitable curvature positivity. In this paper, we prove that every projective manifold $X$ obtained from a projective toric manifold by a finite sequence of blow-ups at points admits a Kähler metric with $HSC>0$. This statement applies to all rational surfaces and therefore completes Hitchin's result, resolving the complex surface case of a problem of Yau listed in "Open Problems in Geometry". The proof has two main ingredients. First, we prove that the toric Kähler metric on a projective toric manifold arising from Delzant's construction has $HSC>0$. Second, via a one-parameter degeneration, we construct, for any such $X$, a smooth projective family $\pi:\mathcal X\to\mathbb C$ such that $\mathcal X_t\simeq X$ for $t\ne0$, while $\mathcal X_0$ is a projective toric manifold.

Two families $\mathcal{A}, \mathcal{B} \subset \binom{[n]}{k}$ are cross-intersecting if $A \cap B \ne \emptyset$ for all $A \in \mathcal{A}$ and $B \in \mathcal{B}$, and non-trivial if neither $\m A$ nor $\m B$ is a star. Pyber proved that any two cross-intersecting families $\mathcal{A}, \mathcal{B} \subset \binom{[n]}{k}$ satisfy $|\mathcal{A}||\mathcal{B}| \le \binom{n-1}{k-1}^2$, and the maximum is attained by two full stars. Frankl, as well as Frankl and Wang, conjectured that the sharp bound, when both families are required to be non-trivial, is $h(n,k)^2$, where $h(n,k) = \binom{n-1}{k-1} - \binom{n-k-1}{k-1} + 1$, the size of the Hilton–Milner family. The cases $k=3$, and the range $k\ge8$ and $n\ge 4k$, were established earlier by Frankl and by Frankl and Wang, respectively. In this paper, we prove their conjecture in the full range. We show that every non-trivial cross-intersecting pair $\mathcal{A}, \mathcal{B} \subset \binom{[n]}{k}$ with $n \ge 2k$ and $k \ge 3$ satisfies $|\mathcal{A}||\mathcal{B}| \le h(n,k)^2$. Moreover, we characterize all extremal pairs. Whereas the corresponding sum problem admits asymmetric and unbalanced extremizers, the product extremum forces a balanced, symmetric-or-dual structure: the two families are isomorphic when $n>2k$ and complement-dual when $n=2k$. Independently and contemporaneously with the present work, Frankl and Wang obtained the same bound for $k\ge8$ and $n\ge2k+1$ by a different method. Our proof combines a diversity technique with several new properties of an extended shift operation. Moreover, we show that the problem behaves differently for different uniformities, exhibiting new extremal configurations. In particular, we disprove a related conjecture proposed by Frankl and Wang.

For additively split stiff evolution problems, classical fourth-order IMEX Runge-Kutta methods usually require several stages and complicated coupling order conditions. This paper proposes a compact fourth-order IMEX-type method based on a two-derivative formulation and analyzes its accuracy and stability properties. The proposed scheme incorporates the mixed explicit-implicit interaction directly through temporal derivatives evaluated along the full vector field, which ensures mixed compatibility for non-commuting split systems. With only one intermediate stage and two implicit solves per time step, the method achieves fourth-order accuracy while improving the accuracy obtained per implicit solve compared with classical multi-stage fourth-order IMEX-RK schemes. In addition, the method exhibits stronger damping of stiff modes in the purely implicit scalar limit and in the strong implicit-stiffness limit with a fixed explicit component. In the purely implicit scalar limit, its stability factor decays quadratically as the stiffness increases, whereas a representative classical fourth-order IMEX-RK reference method shows only linear decay. Numerical experiments on non-commuting split systems, scalar stiff-mode damping, increasing stiffness tests, and one- and two-dimensional advection-diffusion high-mode problems confirm the mixed consistency, the predicted stiff decay, and the smaller errors obtained under equal implicit-solve budgets in strongly stiff regimes.

We study discrete Schrödinger operators on \(\ell^2(\mathbb Z)\) generated by the vector-frequency two-step skew shift \(T_\omega(x,y)=(x+\omega,x+y)\), with an arbitrary nonconstant real-analytic sampling function on \(\mathbb T^d\times\mathbb T^d\). For frequencies in a fixed vector Diophantine class and sufficiently large coupling, we prove stretched-exponential large-deviation estimates at every sufficiently large integer scale, uniformly in the real energy and in the frequency. We also obtain positivity of the Lyapunov exponent for all real energies and a quantitative local continuity estimate in the energy. The proof addresses the simultaneous losses caused by the \(n^{-1}\) complex width induced by the shear, vector Diophantine small divisors in quadratic Weyl sums, and several-variable subharmonic concentration. The main ingredients are a vector Weyl estimate with Poisson regularization, a short-block large-deviation estimate on anisotropic shrinking strips, a several-variable boosting argument, and an all-offset avalanche comparison followed by strong induction.

We give a short proof of the Real Snaith equivalences and multiplicative refinements thereof. The key ingredient is control over structured Real orientations, which we manage through Wilson space theory. In particular, we develop a theory that produces $\mathbb{E}_6$-complex orientations of even periodic $\mathbb{E}_{\infty}$-ring spectra. This machinery can be used to recover an $\mathbb{E}_{2\rho}$-algebra structure on Real Brown-Peterson theory. We apply the Real Snaith theorems to compute $\mathrm{THR}(\mathrm{KU}_{\mathbb{R}})$ and $\mathrm{THR}(\mathrm{MUP}_{\mathbb{R}})$. This requires a norm inverted variant of the Real Snaith theorems, which we prove via the nilpotence theorem.

Lagrangian systems subject to fractional damping can be incorporated into a variational framework by doubling the state variables and introducing fractional derivatives. Fractional variational integrators based on backward-differentiation convolution quadrature (BD-FCQ), combined with higher-order Galerkin methods, saturate at second-order accuracy because the multistep structure of BDFCQ does not take into account the internal stages of the Galerkin discretization. The main objective of this paper is to develop fractional variational integrators (FVIs) by combining Runge-Kutta convolution quadrature (RKCQ) for the approximation of fractional derivatives with higher-order Galerkin methods. The RKCQ approach is naturally compatible with such stage-based discretizations and is therefore better suited for the construction of higher-order schemes. We are particularly interested in the CQ based on Lobatto IIIC. Preservation properties such as energy decay, as well as convergence properties, are investigated numerically and proved for second-order schemes. The presented schemes reach 2nd, 4th and 6th order of accuracy. A brief discussion on the midpoint fractional integrator is also included.

We classify torsion pairs in an essentially small abelian category through cosilting subsets of the Ziegler spectrum of the ind-completion of the abelian category. For Artin algebras, this classification is reformulated as an infinite analog of $\tau$-tilting theory, where torsion classes correspond to support $\tau$-tilting subsets of the Ziegler spectrum and torsion-free classes correspond to support $\tau^-$-tilting subsets. We further express the classification through ideals of the module category, thereby obtaining a formulation that involves finite length modules only. The developed theory is applied to study generic bricks and generic $\tau^-$-rigid modules, in particular for tame algebras, for which we show that these classes of modules coincide. We also recover a result of Bautista, Pérez and Salmerón stating that a tame algebra admits infinitely many bricks of a fixed dimension if and only if there exists a generic brick. Finally, we prove that every algebra whose Krull-Gabriel dimension is defined satisfies the brick version of the second Brauer-Thrall conjecture.

In this paper we prove some concentration inequalities for two types of error probabilities in the Empirical Risk Principle (ERP) in statistical learning, which provide a lower bound and an upper bound for the minimal risk (in terms of the minimal empirical risk) with non-asymptotic high confidence. The usual boundedness condition of the empirical risk function is relaxed to the Gaussian or exponential integrability condition. The confidence of the lower bound of the minimal risk is shown to be independent of the number of training parameters and the dimension of the input vectors, allowing one to detect the deficiency of a learning machine efficiently; and the confidence of the upper bound of the minimal risk is proved to be high provided that the sample size $n$ is much greater than the box dimension of the parameter set $\Theta$ in the Orlicz metric $d_{\psi_1}$ associated with the risk functions. Our work is based on Talagrand's concentration inequalities (the sharp versions by Bousquet and Klein-Rio), transport-entropy inequalities and the recent progress in the theory of empirical processes and statistical learning.