Papers

The Discrete Kirchhoff Triangle (DKT) method for the biharmonic equation is analyzed in the discrete energy norm. The error is bounded by the best approximation of the Hessian by piecewise constants and the oscillation of the right-hand side, without additional regularity assumptions on the exact solution. This result implies first-order convergence of the classical DKT element and the analysis yields a canonical extension to three space dimensions with the same approximation properties. Residual-based a posteriori error estimates are derived. The analysis is formulated within a general framework for low-order nonconforming methods, which also applies to various classical elements and yields best-approximation results by constants. It is furthermore shown how known stable pairs for the planar Stokes system have discrete stream functions in discrete Kirchhoff spaces. This yields variants of the known schemes with positive definite formulations and pressure-robust error bounds.

Phase unwrapping is an essential preprocessing step for phase-based MRI applications, including susceptibility mapping, field mapping, thermometry, and MR elastography. We present Iterative Laplace-Based Phase Unwrapping (ILPU), a bi-level optimization algorithm. In this method, a lower-level solver recovers a continuous phase increment from an incremental Poisson equation using the discrete cosine transform (DCT), while an upper-level solver refines an integer offset map through quality-guided spatial regularization and a restricted local search. This coupling enables robust unwrapping in low-SNR regions through adaptive smoothness penalties and quality-weighted regularization. We evaluated ILPU on 2D and 3D brain MRI phase images against manually unwrapped reference data, using standard Laplace unwrapping, Flynn, and SEGUE as comparison methods. In 2D, ILPU achieves accuracy comparable to SEGUE. In 3D, ILPU attains a relative error of 2.12% compared with 67.59% for SEGUE and 81.02% for Laplace, demonstrating a clear advantage in volumetric unwrapping. The algorithm has O(N log N) complexity per iteration through DCT-based Laplacian estimation and is numerically faster than both Flynn and SEGUE while preserving superior accuracy. These results indicate that the bi-level optimization framework provides a robust and computationally efficient solution for phase unwrapping in MRI.

In their recent paper, Bergfalk and Smythe prove that the isometry equivalence relation on hyperbolic surfaces with finitely-generated fundamental group is concretely classifiable, and ask whether the same result holds true for 2-dimensional hyperbolic orbifolds, or equivalently, whether the action of $\text{PSL}_2(\mathbb{R})$ on its space of finitely-generated discrete subgroups is concretely classifiable. In this note we answer this question in the affirmative. We then use the result to prove that a nonsingular ergodic $\text{PSL}_2(\mathbb{R})$-space with nonelementary finitely-generated stabilizers is homogeneous, in similarity with a result of Stuck-Zimmer for lattices in semisimple lie groups. The main ingredients of our proof are Selberg's lemma and a result of Greenberg on commensurators.

We develop a discrete-time renewal framework in which renewal events evolve along multiple time coordinates and the sojourn mechanism is described by a general distribution on the multi-index lattice. The resulting processes, called multi-time renewal chains, are studied through multi-index convolution and the associated algebra of multivariate formal power series. This algebraic formulation gives explicit representations for multi-time renewal equations, constructive coefficient formulas, and practical inversion schemes. For computation, we combine FFT-based multidimensional convolution with Newton-type reciprocal iteration to evaluate renewal quantities on large grids. For asymptotics, we prove strong laws and central limit theorems under proportional growth of the observation horizon, including a general central limit theorem for additive functionals and a Gaussian limit for the renewal counting process in directions with a unique rate-determining coordinate. We also study fixed-horizon observations: the terminal age vector induces a genuinely multivariate right-censoring mechanism, leading to an exact nonparametric maximum likelihood estimator and its asymptotic normality. Applications include a binomial–multiset identity, two-attribute warranty evaluation, alternating-renewal availability computation, and discretization-based approximations of continuous-time bivariate renewal and availability models.

We study arrangements of smooth conics and lines in the complex projective plane whose singularities are limited to nodes, tacnodes, and ordinary triple points. The first part of the paper gives numerical restrictions for plus-one generated conic arrangements with defect $\nu(C)=3$ and explains how these restrictions interact with Bézout's theorem, the Dimca–Sernesi bound for the minimal degree of a Jacobian syzygy, and Hirzebruch-type inequalities. In particular, the possible numbers of conics are bounded, and the exceptional low-degree cases are separated from those that remain open. The second part concerns arrangements of total degree at most $6$. We identify the weak and strong Ziegler pairs occurring in the database recorded in the Appendix.

The goal of this note is twofold. First, we provide explicit examples of periodic (though not necessarily lattice) sets that give rise to Gabor systems failing to form frames. Our constructions depend only on the parity of the window function $g$. Second, for a wide range of finite-dimensional function spaces $V$ we show that $V$ contains a function $g$ such that a lattice of high density fails to generate a Gabor frame. In particular, we prove that the Gröchenig-Lyubarskii theorem is sharp in the finite-dimensional space of polynomials with Gaussian weight. More precisely, for every $N\in\mathbb{N}$ and every $\alpha,\beta>0$ satisfying $\alpha\beta=\frac{1}{N+1}$, we give an explicit algorithm for finding an even or odd polynomial $p$ of degree at most $N$ such that $\mathcal{G}(p(x)e^{-\pi x^2}, \alpha\mathbb{Z} \times \beta\mathbb{Z})$ does not form a frame. The proofs are constructive, elementary, and based on linear algebra.

In this note we prove a conjecture by Hasegawa stating that a simply connected, nilpotent Lie group of dimension $2n$ endowed with a left-invariant complex structure is biholomorphic to $\mathbb{C}^n$.

This work presents a comprehensive analysis of coupled stochastic van der Pol oscillators, a paradigm for understanding synchronization, bifurcations, and chaos in nonlinear systems subject to random fluctuations. The system comprises two or more oscillators with nonlinear damping, linear diffusive coupling, and additive Gaussian white noise. We develop a unified framework that systematically connects global bifurcations, synchronization phenomena, and chaotic dynamics within a single coherent stochastic model. We explore the stochastic dynamics of coupled van der Pol oscillators by seamlessly blending theoretical principles with in-depth numerical simulations. This integrated approach forms a robust framework for analysis, with essential phenomena clearly depicted in the accompanying figures. We then extend this framework to a comprehensive investigation of large networks, focusing on their continuum limit, emergent pattern formation, the role of noise, and the onset of collective chaos.

We introduce the average confidence width $\Delta_{a} x=\int_0^1\Delta_{c} x (\theta)\,\mathrm{d}\theta$: the confidence width $\Delta_{c} x(\theta)$—the measure of the smallest region of position space carrying probability $\theta$—averaged over all confidence levels. It is the first moment of the decreasing rearrangement of the position density, an $L^1$ (mean-absolute-deviation) measure of localization, and $\Delta_{a} x\,\Delta_{a} p$ is dilation invariant. We study the uncertainty relation $\Delta_{a} x\,\Delta_{a} p\ge c^\ast\hbar$ for pure and mixed states. A mean–entropy argument with the Bialynicki-Birula–Mycielski relation gives $c^\ast\ge\pi/e$, while the ground state $\psi_0$ of the Fourier-invariant operator $|x|+|p|$, with $E_0=1.1040744$, gives $c^\ast\le E_0^2\approx1.2190<4/\pi$: Gaussian states, which saturate the Heisenberg–Kennard and entropic relations, are not optimal here. We prove that $\psi_0$ is nondegenerate, nodeless, even, self-dual, and symmetric decreasing, with algebraic tails, and that the sharp hybrid relations $\Delta_{a} x \cdot 2 \langle |p-p_0| \rangle \ge E_0^2\hbar$ and $2\langle|x-x_0| \rangle \cdot\Delta_{a} p\ge E_0^2\hbar$ hold for all states, so $\Delta_{a} x\,\Delta_{a} p\ge E_0^2\hbar$ whenever either density is symmetric unimodal; the second variation of $\Delta_{a} x\,\Delta_{a} p$ at $\psi_0$ is positive definite apart from the symmetry directions. Extensive numerical minimization, including coherent states, never finds a product below $E_0^2\hbar$ and consistently returns $\psi_0$, supporting the conjecture $c^\ast=E_0^2$. In two dimensions the relation is sharp and saturated by Gaussians, identifying the dimension as the origin of the non-Gaussian optimizer in one dimension. Finally, we discuss other complementary pairs and the relation of the width to interferometric predictability and visibility.

In this article, assuming Artin's (holomorphy) conjecture, we establish an explicit asymptotic formula for the number of non-trivial zeros, up to any given height $T\geq 1$, of Artin $L$-functions. As a consequence, our result yields an unconditional explicit zero-counting formula for Hecke $L$-functions over any number field. In addition, our result improves the recent work of Amberger on Dedekind and Riemann zeta functions and the previous work of Bennett-Martin-O'Bryant-Rechnitzer on Dirichlet $L$-functions for sufficiently large $T$.

In this paper, we investigate the rate of convergence of the relative value iteration (RVI) algorithms for diffusions in $\mathbb{R}^d$ under both the conventional ergodic cost (CEC) and ergodic risk-sensitive cost (ERSC) criteria, and under the uniform exponential stability condition. The existing RVI algorithms for the CEC and ERSC problems solve the associated initial value Hamilton-Jacobi-Bellman type equations whose solutions are shown to converge asymptotically to the corresponding optimal values. However, the rates of convergence for such algorithms have remained open. This paper proposes discrete-time implementations for the RVI algorithms based on slight modifications of the associated PDEs, and proves that the rates of convergence of these RVI algorithms are exponential under a weighted sup-norm. These implementations have discrete-time iterates that can be explicitly expressed as recursive systems. The difference between these iterates and the desired value function in the CEC case can then be expressed in terms of the associated Markov kernels. Similarly, this can be done for the logarithms of the corresponding iterates and desired value function in the ERSC case in terms of the associated Markov kernels for the extended diffusion. As a result, we are able to prove the desirable contraction properties in order to establish the exponential rate of convergence by making use of a weighted semi-norm in which Markov kernel acts a contraction.

We introduce a geometric formulation of statistical feature learning for supervised regression. Feature learning is defined through a base–fiber decomposition: the base is the feature-side geometry produced by training, and the fiber is the learned feature space where estimation is performed. We prove this property for spherical mean-field Langevin dynamics, viewed as the Wasserstein gradient flow of a negative entropy-regularized empirical risk. In Gaussian multi-index models, the low-temperature stationary distribution concentrates near the hidden indices, forms a multi-spike structure, and yields parameter recovery with high probability, even though negative entropy regularization penalizes concentration. This concentration has a sharp transition at temperature $\lambda\asymp 1$. In Gaussian single-index models, the stationary measure satisfies a concentration property, with parity determining whether it lives on $S_2^{d-1}$ or $\mathbb{RP}^{d-1}$. The induced learned feature space aligns the regression signal and yields rates $d/N$ and $Md/N$, up to logarithmic factors.

In both classical and quantum error correction, heralded erasures are known to be easier to tolerate than unheralded general stochastic errors. Whilst an established benefit of loss-dominant quantum architectures such as photonic qubits, this fact has received renewed interest, with a pivot towards reconstructing other architectures to be erasure-dominant, such as dual-rail transmons. This work investigates exploiting these 'erasure qubits' in the near term by using postselection as a technique for error mitigation, wherein circuit shots detecting any erased qubits are discarded from the computational ensemble and repeated. Firstly, we outline a numerical framework for representing circuit-level erasure noise and present 'erado', an open-source library capable of simulating erasure noise and postselection. Secondly, we investigate the effects of both erasure noise and noise in the erasure checks themselves on the quantum Fourier transform (QFT), in the additional presence of gate depolarising noise. A worked example is provided of postselection fully mitigating against the erasure channel for erasure check error rates less than 3.0%. We also show how a postselected dual-rail system can surpass a fundamental noise floor at the kiloquop scale where a comparable single-rail system cannot, justifying this approach in the NISQ regime before (and, perhaps, combined with) the practical arrival of QEC.

This paper proposes a modular vehicle system for passenger-freight integration along a bidirectional transit corridor. The system uses homogeneous units that can be coupled into vehicles and assigned to either passenger or freight service. Freight is carried by dedicated units, with loading and unloading coordinated with docking and undocking and separated from passenger boarding and alighting. To better respond to uncertain passenger demand and integrate freight transport, vehicles can be reconfigured at intermediate stations, where they can also depart and terminate. We jointly optimize departure-specific service routes, timetables, vehicle compositions, unit schedules, and passenger-freight demand assignments, with unit reuse constrained by explicit docking and undocking times. These decisions are modeled on a space-time-state network and formulated as a stochastic mixed-integer program that minimizes unit deployment costs, passenger waiting costs, and penalties for unmet freight demand. Passenger demand uncertainty is addressed using linearized chance constraints. To solve the problem, we develop an exact Benders decomposition algorithm with valid inequalities and a warm-start strategy, together with a tailored decomposition-based heuristic for larger instances. Computational experiments on instances generated from representative transit corridors in Gothenburg demonstrate the effectiveness of the Benders algorithm for small- and medium-sized instances and the scalability of the heuristic for larger problems. Sensitivity analyses highlight the value of accounting for passenger demand uncertainty and the effects of temporal overlap between passenger and freight demand. Comparisons with benchmark transit systems further demonstrate the operational advantages of the proposed modular integrated system.

We establish a coordinate-free differential geometric framework for anomalous transport in topological bands using the Hodge-de Rham decomposition of the Brillouin zone. Standard formulations face mathematical singularities (Dirac strings) when using the quantum Berry connection in bands with non-zero Chern numbers. Applying this decomposition to the Berry curvature 2-form isolates the quantized topological monopole flux from a globally smooth geometric 1-form proxy potential, $\mathcal{A}$. Substituting this regularized potential into semiclassical transport integrals yields distinct analytical advantages. For linear transverse transport, our cohomological decomposition enables an exact geometric derivation of Haldane's insight via the co-area formula, partitioning the response into a continuous Fermi sea topological background and a localized Fermi surface geometric line integral. For non-linear transport, this globally smooth proxy unifies the geometric description, reproducing the high numerical stability of scalar integration-by-parts techniques directly from its exact sector, accommodating arbitrary Chern numbers. By enforcing the continuous Coulomb-Hodge gauge ($\delta \mathcal{A} = 0$) alongside vanishing harmonic holonomies over fundamental 1-cycles ($\oint_{\gamma_i} \mathcal{A} = 0$), we map the Hodge potential $\mathcal{A}$ to the Maximally Localized Wannier Function (MLWF) gauge in trivial bands, providing a non-singular computational proxy for topologically obstructed bands. Finally, we analytically demonstrate that solving the Hodge Laplacian for $\mathcal{A}$ zeroes the macroscopic Brillouin zone average (uniform $\mathbf{R}=0$ zero-mode) topological divergence, yielding a mathematically consistent covariant formulation that matches the algorithmic robustness of standard methods against discrete $\mathbf{k}$-grid noise.

This paper focuses on the analysis of an initial-boundary value (direct) problem for the Hallaire-Luikov moisture transfer equation involving the $\psi$-Prabhakar integral-differential operator of fractional order. We establish the existence, uniqueness, and stability of the solution to the formulated problem. To construct the solution, we employ the method of separation of variables and the method of successive approximations (iteration method), and obtain the solution to the considered problem in an explicit form. Furthermore, the solution is expressed in terms of a novel quadrivariate Mittag-Leffler-type function. An a priori estimate for the problem is also established.

Let $k$ be a non-archimedean local field of characteristic zero. We give sufficient conditions under which the Ginzburg-Rallis models of the induced representations of $\mathrm{GL}_6(k)$ from a parabolic subgroup of type $[2^3]$ are isomorphic to the trilinear models of the inducing data. We also give nonvanishing criterion for these trilinear models and Ginzburg-Rallis models.

We study the image of a regular unipotent element under any finite-dimensional irreducible polynomial representations of $\mathrm{GL}_3(\mathbb{C})$. This problem is equivalent to decomposing certain compositions of irreducible representations as $\mathrm{SL}_2(\mathbb{C})$-modules. We give an explicit decomposition of this finding, its Jordan decomposition.

We construct holomorphic differential forms of many degrees, including the minimum possible one, on the modular varieties associated to the even lattices of signature $(2, n)$ with $n\equiv 1, 3$ mod $8$ and discriminant $-2$ in the range $n\geq 25$. This is the first example of holomorphic differential forms of non-top degree on orthogonal modular varieties. The proof uses the Arthur multiplicity formula in the theory of automorphic representations.

We relate the hyperbolicity of a calibrated manifold $(X, \phi)$ to the analytic properties of the space of Smith immersions $\mathrm{SmIm}(B^k, X)$ from the Poincare $k$-ball into $X$. In particular, we establish the following calibrated analogue of a theorem of Royden: if $X$ is $\phi$-saturated, then $R_\phi$- and $K_\phi$-hyperbolicity coincide, and either implies the equicontinuity of $\mathrm{SmIm}(B^k, X)$ with respect to the $\phi$-distance. This yields a Montel theorem for compact $\phi$-saturated calibrated manifolds as an immediate corollary. Our primary technical tool is a new Schwarz lemma for Smith immersions from $B^k$ into $X$, which is of independent interest. In a similar spirit, we also prove a calibrated analogue of Kiernan's theorem to the effect that the $K_\phi$-hyperbolicity of $X$ is almost equivalent to $\mathrm{SmIm}(B^k, X)$ being a normal family. Finally, we prove that bounded domains in flat euclidean space are $R_\phi$-hyperbolic for any calibration $\phi$, and we investigate the hyperbolicity of products and discrete quotients.

In the siblings version of the coupon collector, a main collector stops when every coupon type has appeared once. Duplicates are passed successively to siblings, and $U_j^N$ denotes the number of empty spaces in the $j$th collector's album at the main completion time. We prove finite-$N$ radial transform strengthenings of the uniform-probability extremality principle. For every $N\ge2$, every $j\ge2$, every positive nonuniform probability vector $p$, and the ray $p(\theta)=u+\theta(p-u)$ from the uniform vector $u$, the full probability generating function $\mathbb{E}_{p(\theta)}z^{U_j^N}$ is strictly decreasing in $\theta$ for $z>1$ and strictly increasing in $\theta$ for $0<z<1$. Thus the same full PGF has opposite radial monotonicity on the two sides of $z=1$, the left side giving a radial Laplace-transform order. At the coefficient level, along every nonconstant ray from the uniform vector, uniform probabilities maximize every binomial moment of $U_j^N$, equivalently giving a finite absolutely-monotone/binomial-transform order. The proof of the right-PGF and binomial-moment theorem is exact and finite-dimensional. It uses Poissonization, a marked Poissonized PGF identity, a normalized alternating subset expansion, and a positive-kernel radial derivative formula obtained from a local cumulative-polynomial dissipation lemma. The Laplace-transform theorem follows from a separate Gamma-mixture race representation.

Low-rank matrix optimization is often carried out via the Burer-Monteiro (BM) formulation, but choosing the factorization rank $r$ is delicate and can substantially slow optimization. We propose a unified framework, termed direction-magnitude decomposition (DMD), that decomposes the optimization variable to improve optimization efficiency even when the target rank is unknown. We develop two DMD-based approaches and establish their theoretical advantages on the canonical problem of matrix factorization. The first, overparameterized DMD, uses a rank $r$ larger than necessary and enjoys faster convergence as $r$ increases. The second, recursive DMD, is motivated by the incremental eigenpair learning, or saddle-to-saddle, behavior of overparameterized DMD. It achieves lower memory and computational costs, complementing overparameterized DMD. Both approaches are exponentially faster than gradient descent applied to the BM formulation. Numerical experiments on matrix factorization, sensing, and completion corroborate our theoretical findings and demonstrate the practical effectiveness of DMD.

Acoustic scattering arises in a wide range of applications, including medical imaging, geophysical exploration, acoustic metamaterials, etc. In this paper, we develop a fast and highly accurate algorithm for acoustic scattering by multiple quasi-axisymmetric objects, whose axis of rotation is an arbitrary curve. The method is based on a Nyström discretization that combines Gauss-Legendre quadrature with the trapezoidal rule. To treat the singular integrals that occur when target points are close to or coincide with source points, we reformulate them as evaluations of the modal Green's function and its derivatives, which are computed efficiently using the fast Fourier transform and convolution. The multiple scattering solver is then constructed by coupling the single scatterer discretizations through inter-body boundary integral interactions. We also present a convergence analysis for scattering problems with smooth geometries. Numerical examples demonstrate the efficiency and accuracy of the proposed method for solving multiple scattering problems involving up to 1000 quasi-axisymmetric structures.

A rainbow stacking of $m$ independent, uniformly random $r$-edge-colourings of $K_n$ is a tuple of vertex permutations that superimposes the colourings such that no two edges of the same colour overlap. The study of the critical palette size $r$ required for the existence of such stackings was recently initiated by Alon, Defant, and Kravitz [Bull. Lond. Math. Soc., 57, 2025], who bounded the phase transition within a constant-order window around $\frac{m\binom{n}{2}}{2\log(n!)}$. We determine the constant term in this transition. For every fixed $m\ge2$ and every function $\omega(n)\to\infty$, with high probability there is no rainbow stacking if $$r\le \frac{m\binom{n}{2}}{2\log(n!)}+\frac{2m-1}{6}-\frac{\omega(n)}{(\log n)^2},$$ while with high probability there is one if $$r\ge \frac{m\binom{n}{2}}{2\log(n!)}+\frac{2m-1}{6}+\frac{\omega(n)}{(\log n)^2}.$$ Our proof combines a chromatic-polynomial expansion for an auxiliary conflict graph with a refined estimate of the associated weighted permutation sum. Our result yields the exact threshold $\Big\lceil \frac{m\binom{n}{2}}{2\log(n!)}+\frac{2m-1}{6}\Big\rceil$ for a density-one set of integers $n$, resolving a problem of Alon, Defant and Kravitz.

In this article, we study bilinear Calderón–Zygmund operators on a Vilenkin group $G$. As a preliminary step, we establish a Grafakos–Torres-type endpoint weak-type result in our setting. Furthermore, we prove that such operators extend to bounded bilinear mappings from $L^{p_1}(G)\times L^{p_2}(G)$ into $L^p(G)$ under the natural condition $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}.$ We then obtain a corresponding boundedness result in Morrey spaces, showing that these operators extend to bounded bilinear mappings from $\mathcal{M}_{p_1,u_1}(G)\times \mathcal{M}_{p_2,u_2}(G)$ into $\mathcal{M}_{p,u}(G)$ under suitable assumptions. These results generalize the classical bilinear estimates to the setting of Vilenkin groups.

We provide an explicit construction of a Gabor orthonormal bases for a local field $K$ that provides maximal localization in both time and frequency. Such a localization is not true in case of $\mathbb{R}$ due to the uncertainty principle. In particular, we construct examples of functions $f \in L^2(K)$ such that the support of the ambiguity function of $f$ is of minimum measure. Moreover, we establish a quantitative uncertainty principle for local fields, which follows as a consequence of Lieb's inequalities for general locally compact abelian group. In addition, we develop fundamental operator representations for Gabor systems defined over local fields.

Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error. Truncation-based methods often require problem-dependent artificial boundary conditions, while global spectral bases may be inefficient for localized structures, irregular geometries, or solutions with different near-field and far-field behaviors. We propose a domain-decomposed randomized neural network framework for such problems. Different randomized subnetworks are assigned to different spatial regimes: a near-field subnetwork captures local and geometric features, whereas a far-field subnetwork represents exterior decay. The subnetworks are coupled by boundary and interface conditions, and only the output-layer coefficients are solved from linear least-squares systems arising from Petrov–Galerkin or collocation formulations. We develop a Petrov–Galerkin method for semi-unbounded elliptic problems and a collocation method for fully unbounded, perforated, and time-dependent problems. A conditional bounded-parameter approximation result is proved in a broken Sobolev norm, together with an error decomposition covering approximation, empirical-consistency/quadrature, and least-squares optimization errors. Numerical experiments for Poisson and time-dependent Schrödinger equations demonstrate the accuracy and flexibility of the proposed method.

Let $p$ be a prime number, and let $k$ be a finite field of characteristic different from $p$. Let $X$ be a normal geometrically connected variety over $k$, let $\overline X$ be a compactification of $X$, and let $Z=\overline X\setminus X$. Let $D$ be an effective Cartier divisor on $\overline X$ whose support is contained in $Z$. Motivated by Hiranouchi's Hermite–Minkowski type theorem for varieties over finite fields, we formulate a finiteness conjecture for continuous semisimple geometric representations $$ \pi_1(X,D)\longrightarrow \operatorname{GL}_n(F), $$ where $\pi_1(X,D)$ is Hiranouchi's fundamental group with ramification bounded by $D$, and $F$ is an algebraically closed field of characteristic $p$ endowed with the discrete topology. We prove this conjecture for odd $p$ in the following two cases: for curves with arbitrary ramification bound $D$, and for varieties of arbitrary dimension in the tame case, namely $D=0$. Furthermore, for arbitrary $p$, we prove the finiteness for those representations which admit a lift to characteristic zero.

The aim of this paper is to show that the spectral theory based on the S-spectrum is particularly well suited for the Dirac operator on manifolds, even in cases where the operator is not self adjoint. Traditionally, for non-self adjoint operators in the Clifford setting, the literature has often referred to the right spectrum. However, a more comprehensive approach is provided by the theory of the $S$-spectrum, which is the appropriate notion for general operators on Clifford modules. In this work, we show that this theory is particularly well suited for bisectorial Clifford operators. By using the $S$-spectrum, which naturally contains the right eigenvalues, we prove bisectorial estimates for the $S$-resolvent associated with the spinor Dirac operator under various boundary conditions.

We consider the analytic index and spectral flow of Fredholm operators on Hilbert $C^*$-modules. Our spaces and algebras are equipped with a real structure, so the analytic index and spectral flow takes value in the real $K$-theory group of a $\sigma$-unital $C^*$-algebra. We use Van Daele $K$-theory, which allows us to treat the eight real $K$-theory groups and the two complex groups on an equal footing. We provide a general definition of the analytic index for Clifford anti-linear and skew-adjoint Fredholm operators as well as self-adjoint and odd Fredholm operators. Our definition of spectral flow and its basic properties are valid for Wahl-continuous paths of Fredholm operators on a real Hilbert $C^*$-module. We also provide an analytic approach to the spectral flow as a decomposition into a finite sum of relative indices. Furthermore, we prove a real version of the Robbin-Salamon theorem, relating the spectral flow to a Fredholm index. Our description of the index and spectral flow relies on various isomorphisms between Kasparov's $KKR$-theory and Van Daele $K$-theory, which we systematically describe in the Appendix.