Papers

We study the graph Laplacian operator as a regularizer in a generalized Tikhonov framework for linear ill-posed problems. The Laplacian is updated iteratively from the current reconstruction, so that progressively sharper structural information about the solution is fed into the regularization term. We introduce three schemes: a standard one that rebuilds the Laplacian from each new iterate; an error-equation scheme that, following the error-based formulation of iterated Tikhonov regularization, builds the Laplacian from an estimate of the reconstruction error rather than of the image itself; and a mixed scheme combining the two. We establish convergence of all three schemes for noisy data under a priori parameter and stopping rules as the noise level tends to zero. Numerical experiments in two-dimensional computed tomography and image deblurring show consistent gains in reconstruction quality and sharper recovery of fine details.

We establish existence, uniqueness, and local \(L^1\)-stability for multidimensional scalar conservation laws with discontinuous heterogeneous flux, without imposing a non-degeneracy condition on the physical space–time flux. The discontinuity set may be a locally finite family of \(C^2\) hypersurfaces whose intersections and singular points are contained in a closed set of vanishing \((d-1)\)-dimensional Hausdorff measure. The main obstacle is the possible failure of multidimensional compactness when a nontrivial linear combination of the space–time flux components is constant on an interval of states. We show that, for interface problems, this obstruction can be reduced to the physical normal flux. Its flat intervals are collapsed by a normal-flux quotient that preserves the normal flux and hence the Rankine–Hugoniot relation. On the non-flat region, the time and tangential components are replaced by auxiliary polynomial functions of the normal flux, while the physical normal component is left unchanged. The resulting auxiliary space–time vector satisfies the interval non-degeneracy required by Panov's compactness theory. This yields strong compactness and strong one-sided traces of the quotient variables. The interface admissibility condition is obtained by projecting the vanishing-viscosity germ associated with the two one-sided physical normal fluxes. A localized viscous comparison shows that the quotient traces belong to this maximal \(L^1\)-dissipative germ, and a Young-measure contraction argument recovers strong convergence of the physical states. Curved interfaces are treated by flattening, localization, and finite-propagation patching. The theory is first constructed for \(BV\) initial data and then extended by continuity to \(L^1_{\mathrm{loc}}\cap L^\infty\) data taking values in the invariant interval.

A graph is matchable if it admits a perfect matching. Recently, Goedgebeur et al. asked whether there exists a constant $c<12$ such that infinitely many matchable planar $3$-connected graphs, each with exactly $c$ perfect matchings. We answer this question by proving that every matchable planar $3$-connected graph on at least 40 vertices has at least 12 perfect matchings, and this lower bound is sharp.

Russell's paradox was proposed in the early 20th century to address loopholes in set theory, which directly triggered the third mathematical crisis. This paper proposes and elaborates a perspective distinct from previous studies: paradoxes do not give rise to contradictions; instead, they constitute a Möbius strip-style self-consistent logical structure via self-reference and negation under the logical rules within a system. Gödel's incompleteness theorems indicate that such structures universally exist in formal logical systems. Turing proved the undecidability of the halting problem by first assuming the existence of a halting program and subsequently refuting this assumption through paradox construction, and this paper demonstrates flaws inherent to such proof strategy. Cases of paradoxes within three-valued logical systems are further discussed in this work, where the undecidability of paradoxes is rigorously proven. Finally, inspirations drawn from paradoxes for the real world are explored: two opposing factors can be integrated through the joint mechanism of self-reference and negation. A representative example is the wave-particle duality of light, whose essence may be interpreted as a paradox of waves and particles.

Highly constrained multi-objective design and decision problems are difficult to solve because of strict feasibility requirements and conflicting stakeholder preferences. Evolutionary algorithms are widely used for these problems but typically separate constraint handling from preference optimisation. Pareto-based methods provide a set of trade-off solutions but require post-processing to select a preferred option, while scalarisation can distort heterogeneous stakeholder preferences. This paper introduces the Integrative Maximisation of Aggregated Preferences (IMAP) and its Inter-Generational Solver (IMAP-IGS), which embed preference aggregation directly into the evolutionary fitness function. A constraint-violation preference function guides the search toward feasibility, while final scores reflect only stakeholder preferences. The framework can be integrated into any evolutionary algorithm that uses scalar fitness evaluation. Two implementations are presented: IMAP-BRKGA for combinatorial optimisation and IMAP-GA-II for continuous optimisation. Testing on DAS-CMOP, MO-VRPTW, and HVASP benchmarks shows that IMAP consistently outperforms weighted-sum and Pareto-based approaches when stakeholder preferences conflict. These advantages largely disappear when preferences align, indicating that IMAP's strength lies in resolving preference conflicts in highly constrained decision environments.

Let $E/\Q$ be an elliptic curve and let $\widetilde K_p$ be the compositum of all $\Z_p$-extensions of a quadratic field $K$. We prove that $E(\widetilde K_p)_{\tors}=E(K)_{\tors}$ for $p\geq5$. For $p=3$ and imaginary quadratic $K\neq\Q(\sqrt{-3})$, torsion on each extension is determined by its intersections with the cyclotomic extension and the $2$-division field. Over $\Q(\sqrt{-3})$, we construct infinitely many non-CM curves with full $3$-torsion in the first anticyclotomic layer and compute the $3$-primary torsion on every slope for eight CM curves. For $p=2$, we bound the odd-primary torsion and exclude all primes greater than $7$. We also give uniform bounds for non-CM primary torsion and correct two assertions in Li's preprint about noncyclotomic $\Z_3$-extensions.

We construct analogues of Nakajima's nilpotent quiver varieties for quivers with multiplicities, by developing new stability conditions for framed nilpotent quiver representations with multiplicities and exploiting their interplay with Hecke correspondences. Using these new quiver varieties, we extend Saito's geometric construction of Kashiwara crystals of irreducible highest-weight representations of Kac–Moody algebras from the symmetric to the symmetrisable case.

Low-dose computed tomography (LDCT) reduces radiation exposure but increases the ill-posedness of the reconstruction problem due to noise and sparse data. While regularized methods like Tikhonov and Total Variation (TV) improve image quality, their performance depends heavily on noise characteristics, sampling conditions, and parameter selection. This study presents a systematic stability and robustness analysis of Filtered Back Projection (FBP), Tikhonov regularization, and TV minimization within a 2D parallel-beam CT framework. A unified simulation pipeline based on the Radon transform is developed and evaluated using both the modified Shepp-Logan phantom and a clinical thorax image. Reconstruction behavior is investigated under multiple degradation scenarios involving Gaussian, Poisson, and mixed noise models, across baseline (180 projections) and sparse-view (60 projections) acquisition geometries. To ensure a fair comparison, regularization parameters are optimized for each scenario through an exhaustive SSIM-based grid-search. Quality is assessed via RMSE, PSNR, and SSIM, while robustness is quantified through an empirical Stability Factor S measuring perturbation amplification from measurement to image space. The results show that FBP is highly sensitive to noise and undersampling. Tikhonov regularization improves structural fidelity compared with FBP but remains more sensitive to perturbation than TV. Conversely, TV provides the best compromise between noise suppression, edge preservation, accuracy, and numerical stability. These findings highlight the stability-resolution trade-off in LDCT and demonstrate that the proposed Stability Factor S offers valuable complementary information to conventional metrics.

Signed enumerations of l-regular partitions by the parity of the number of parts are known to correspond to partitions with congruence conditions. Inspired by the recent combinatorial approaches of Ballantine-Merca and Liu for such identities, we provide combinatorial proofs of three identities for l-regular partitions and their variants. Two of these were originally established analytically by Hickerson and Robbins, respectively.

Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of an $n$-vertex graph is less than or equal to the number of its edges plus $\binom{k+1}{2}$ for every $k\in \{1,2,\dots,n\}$, which has been confirmed by Kothari and Tudose (2026) recently. In this note, we characterize the equality case in this inequality. Our main result is that for every $n$-vertex graph $G=(V,E)$ and for every $k\in \{1,2,\dots,n-1\}$, the equality $\sum_{i=1}^k\mu_i(G)=|E(G)|+\binom{k+1}{2}$ holds if and only if $G$ is a threshold graph with clique number $k+1$, where $\mu_1(G)\geq \mu_2(G)\geq \cdots\geq \mu_{n}(G)$ are the Laplacian eigenvalues of $G$. This, together with the confirmed Brouwer's conjecture, would yield a complete solution to the full Brouwer's conjecture posed by Li and Guo (2022). Our proof relies on the projection method of Kothari and Tudose and shows directly that the equality case can occur only for threshold graphs.

This project investigates the Phoa principle in synthetic domain theory (SDT), and provides a generalisation to the transfinite cases. The Phoa principle plays a pivotal role in SDT by illustrating how the paths give the information order on the interval type and other algebraic structures in SDT. The project defines the dual simplices and spines and introduces the concept of sobriomorphisms, which contributes to a new interpretation of the Phoa principle and its generalisations. Finally, the project proposes a hypothetical completeness theorem that may unify the Segal completeness and the chain completeness in SDT based on investigations on the Phoa principle in the project. The project also includes axiomatisation of the interval type in Cubical Agda and the formalised proof for the main theorems.

Mixed finite element methods based on Raviart–Thomas spaces are widely used for the numerical approximation of second–order elliptic problems in flux form. On quadrilateral meshes, however, the bilinear mapping from the reference element introduces a spatially varying Jacobian, which may violate the inclusion property $\mathrm{div}\,V_h \subset W_h$ for the standard Raviart–Thomas spaces. In this paper we propose a simple modification of the classical Raviart–Thomas elements on quadrilateral meshes. The modification consists of adding geometrically motivated correction terms to the local basis functions in order to compensate for the geometric distortion introduced by the bilinear mapping. The resulting spaces retain the same dimension and degrees of freedom as the classical Raviart–Thomas elements while restoring the compatibility property. We present a general framework for constructing such modified spaces and illustrate the approach by developing modified versions of the lowest order and next–to–lowest order Raviart–Thomas elements. Theoretical analysis establishes optimal approximation properties under the standard shape–regularity assumption for quadrilateral meshes. Numerical experiments on distorted meshes confirm the predicted convergence rates and show that the modified elements yield consistently improved accuracy over the classical Raviart–Thomas formulation as the geometric distortion increases.

Causal subgroup analyses often report a small number of groups summarizing treatment effect heterogeneity, as if that number were a well-defined estimand. Outside genuinely latent class populations, however, a “true” subgroup count is model dependent rather than a population functional. We replace it with a new population estimand, the resolution profile, a functional of the causal feature law giving the fewest groups explaining a prescribed fraction of causal heterogeneity, defined for every population without latent structure. Inference is organized around one cross-fitted Bayesian-bootstrap posterior for a single structured moment process, its scores corrected with influence functions, so that paths, profiles, fixed-resolution summaries, and subgroup effects follow by composition. A uniform conditional Bernstein–von Mises theorem over a loss class containing the nonsmooth quantization losses shows this posterior merges with the efficient Gaussian limit under stated nuisance-rate and margin conditions. Subgroup-number uncertainty is not model selection but threshold nonregularity, the profile being an integer-valued threshold of a continuous path, discontinuous in the law at each knot. At these knots no single-valued selector is locally uniformly consistent over root-$n$ neighborhoods, and the set-valued report obtained by inverting a simultaneous band retains locally uniform validity over exactly the same perturbations. Simulations support the approximations, and an analysis of the MineThatData e-mail experiment illustrates the resolution-indexed report, in which two to three groups summarize the visit response while finer structure falls below a noise-floor diagnostic.

Let $n\ge1$ and $C\subseteq\mathbb{R}^n$ be a closed convex cone with conic intrinsic volumes $v_0(C),\ldots,v_n(C)$. We prove the long-standing log-concavity conjecture for this sequence, in the stronger form \[ v_k(C)^2\ge \rho_k\rho_{n-k}v_{k-1}(C)v_{k+1}(C), \qquad 1\le k\le n-1, \] where, for $l\ge1$, $\rho_l=\frac{l+1}{l}\frac{\omega_{l-1}\omega_{l+1}}{\omega_l^2}>1$ and $\omega_j$ is the volume of the Euclidean unit ball in $\mathbb{R}^j$. The proof applies the Alexandrov–Fenchel inequality to the Takemura–Kuriki identity \[ V(A[k],D[n-k])=\frac{\omega_k\omega_{n-k}}{\binom nk}v_k(C),\qquad A=C\cap B^n,\quad D=C^\circ\cap B^n, \] where $C^\circ$ is the polar cone, $B^n$ is the Euclidean unit ball, and repeated arguments are indicated by brackets. A Master Steiner argument gives the identity directly for arbitrary closed convex cones, including degenerate ones. We also give a circular-cone counterexample to the standard ultra-log-concavity normalizations.

The paper studies product-space norm minimisation problems involving distances to convex sets. Using standard tools from convex and functional analysis, we establish complete dual necessary and sufficient optimality conditions and show that the entire solution set can be constructed from the dual vectors arising from the optimality conditions at a given solution. As a consequence, we study optimality conditions and solution set descriptions for generalised versions of the Fermat-Torricelli problem, the Chebyshev centre problem, and the p-Fermat-Torricelli problem. Comparisons with existing results are provided whenever applicable. Examples in finite and infinite dimensional spaces equipped with different norms are presented to illustrate the results.

Let $q$ be an odd prime power and put \[ \theta_q=\frac{\varphi(q-1)}{q-1}. \] An element of $\Fq$ is called non-square non-primitive, or NSNP, if it is both a non-square and a non-primitive element. Let $\ell$ be an odd prime divisor of $q-1$. We obtain a character-sum estimate for the number of translates of an arbitrary finite set lying in the set of non-square $\ell$th powers. Combining this estimate with a finite computation, we prove that $\theta_q<4/15$ guarantees three consecutive NSNP elements. On the boundary $\theta_q=4/15$, the only exceptions are $q\in\{31,61,121\}$. As a further application, we show that for $\operatorname{char}\Fq>3$, the inequality $\theta_q<8/35$ guarantees four consecutive NSNP elements, with $q=211$ as the unique exception on the boundary $\theta_q=8/35$. Thus both constants are best possible.

For every fixed integer $t \geq 2$, we give an upper bound on the multicolor vector space Ramsey number $R_2(t; k)$ that is a tower function of height independent of $k$. For $t \geq 3$, this is the first bound of its form, significantly improving upon the earlier bounds that are towers of height linear in $k$. We achieve this by reducing the problem to a classical hypergraph Ramsey problem via binary simplex codes. In particular, we prove that $$R_2(t; k) \leq \left\lceil \log R(K_s^{(r)}; k + 1) \right\rceil \leq \mathrm{twr}_{r-1}(c k\log k),$$ for $r = 2^{t - 1}$ and $s = 2^t - 1$, where $R(K_{s}^{(r)}; k + 1)$ is the classical $(k + 1)$-color Ramsey number for the complete $r$-uniform hypergraph on $s$ vertices. This improvement also translates into an improved lower bound on the chromatic number of the binary projective space with respect to $(t - 1)$-flats. For $t = 2$, it recovers the connection with multicolor Ramsey numbers for triangles.

Let $X$ be an alphabet, let $t\geq 0$ and $n\geq t+1$, and let $((a_i,b_i))_{i=1}^{m}$ be an ordered family of word pairs in $X^n$ satisfying $dist(a_i,b_i)\geq t+1$ for every $i$ and $dist(a_i,b_j)\leq t$ whenever $i<j$. We prove the sharp bound $m\leq 2^{t+1}$, thereby resolving a problem posed by Alon, Jin, and Sudakov. Our proof uses a linear-algebraic method based on a characteristic-two algebra, which may be of independent interest.

We consider the uniform exponential stability analysis of infinite-dimensional impulsive systems defined on a Banach or Hilbert space, whose flow is governed by a fixed $C_0$-semigroup generator and whose jumps occur at a prescribed time sequence. While the flow and jump maps are themselves time-invariant, the time-triggered impulses render the propagator a genuinely time-varying evolution family, which is the source of the analysis difficulty addressed here. We combine ideas from hybrid systems theory and infinite-dimensional systems to produce operator-based stability conditions, which can be analytically or numerically checked via convex programming. Necessary and sufficient conditions for the uniform exponential stability of impulsive systems on Banach spaces are obtained in the context of a fixed impulse-times sequence but also of arbitrary, constant, minimum, and range dwell-times using both non-coercive and coercive Lyapunov functionals. Some of those results are then adapted to systems on a Hilbert space and quadratic Lyapunov functionals. As an application, linear switched systems are shown to be an exact special case: reformulated as impulsive systems with unit-norm selector jumps, they inherit non-coercive and clock-dependent dwell-time stability conditions on both Banach and Hilbert spaces. Theoretical and numerical examples are given for illustration, notably on the sampled-data control of time-delay systems.

In this paper, we have proved a quantitative version of the approaching geodesic property for certain convex domains. We have proved that that if $\Omega \subset \mathbb{C}^{d}$ is a bounded strongly convex domain with $\mathcal{C}^3$ boundary and $\gamma_{1}, \gamma_{2}:[0, \infty) \to \Omega$ are two geodesic rays such that $\gamma_{1}(\infty)=\gamma_{2}(\infty)=\xi \in \partial \Omega$. Then if the images of $\gamma_{1}$ and $\gamma_{2}$ are contained in the same complex geodesic, then there exists $T\in \mathbb{R}$ \[ \lim_{t \to \infty} \frac{1}{t} \log K_{\Omega}\big(\gamma_{1}(t), \gamma_{2}(t+T)\big) = -2, \] otherwise \[ \lim_{t \to \infty} \frac{1}{t} \log K_{\Omega}\big(\gamma_{1}(t), \gamma_{2}(t+T)\big) = -1. \] Furthermore, using this property we provided a characterization of strongly pseudoconvex domain via a biholomorphic invariant function namely generalized squeezing function. We have proved that: For every $\alpha>0$ there exists $\epsilon(d,\alpha)>0$ such that the following holds: if $\Omega \subset \mathbb{C}^d$ is a bounded convex domain with $\mathcal{C}^{2,\alpha}$-boundary and \[ T_{\Omega}^{D}(z)\geq 1-\epsilon \] outside a compact subset of $\Omega$, where $D \Subset \mathbb{C}^{d}$ is a balanced strongly convex domain with $\mathcal{C}^{3}$ boundary and $T_{\Omega}^{D}$ is the squeezing function of $\Omega$ with respect to the domain $D$ then $\Omega$ is strongly pseudoconvex. We also establish exponential convergence of a certain family of quasi-geodesics in the unit ball of $\mathbb{C}^{d}$. We further show that the study of this family of quasi-geodesics provides a useful tool that allows the exponential convergence property of geodesics to be transferred from local subdomains to the ambient domain, as well as in the reverse direction.

We study the long-time, small-noise behavior of a class of dissipative stochastic evolution equations in a separable Hilbert space, driven by a cylindrical Wiener process whose covariance degenerates to a limiting operator in the strong operator topology. A prototypical example is a stochastic reaction-diffusion equation on a bounded domain with spatially homogeneous but spectrally regularized noise that becomes spatially rough in the limit. We establish a large deviation principle for occupation measures as the time horizon becomes large, the noise intensity tends to zero, and the noise becomes increasingly rough. The result covers a broad class of dissipative equations in infinite dimensions, including those driven by asymptotically rough cylindrical noise whose covariance need not be trace class. Extending the finite-dimensional work of Budhiraja and Zoubouloglou, the infinite-dimensional setting introduces substantial new difficulties: the bound arguments require careful handling of the unbounded evolution and inverse covariance operators, and the increasing roughness of the noise must be balanced against its vanishing amplitude through uniform estimates. Proofs combine analytic semigroup techniques, fractional domain space estimates and a careful treatment of stochastic convolutions in weighted spaces. The rate function is given by a simple explicit formula, the average over the measure of the squared Cameron-Martin cost of canceling the deterministic drift at each point. The proof follows the weak convergence approach based on the Bouè-Dupuis variational formula and constructs near-optimal controls by alternating travel phases that steer the process between prescribed target states and hold phases that stabilize it near a target while shaping the occupation measure.

We consider a mechanical system of infinitely many rotators weakly interacting with each other and prove that the infinite dimensional invariant tori provided by \cite{CGP} are Lagrangian and stable over subexponentially long time.

This is a survey article on the moments of a class of error terms in analytic number theory. We begin by reviewing the moments of the Dirichlet divisor problem, and then review the hybrid moments of error terms in the divisor problems. We give two new results on the joint distribution of sign changes of $\Delta_2(x)$ and $\Delta_3(x). $ In the last section, we present some conjectures on the moment results in the well-known Selberg class.

Let ${\mathbb Z}_{n}$ be the additive group of residue classes modulo $n.$ For any positive integers $n_1$ and $n_2$, let $s(n_1,n_2)$ and $c(n_1,n_2)$ denote the number of subgroups and the number of cyclic subgroups of the group ${\mathbb Z}_{n_1}\times {\mathbb Z}_{n_2}$, respectively. The aim of this paper is to study the asymptotic behavior of the sums $\sum_{n_1,n_2\le x}s(n_1,n_2)$ and $\sum_{n_1,n_2\le x}c(n_1,n_2)$. Some sharper asymptotic results are given for the these sums. Mean values of the error terms are also studied.

In this paper,motivated by the pioneering work of Moslehian and Rassias, we introduce a novel generalized von Neumann-Jordan type constant in Banach spaces, a constant that notably encompasses a range of classical geometric constants. First, we establish the bounds of this constant within Banach spaces. Next, we derive its relationships with other geometric constants. Furthermore, we characterize the uniformly non-square property and provide a sufficient condition for the uniform normal structure by leveraging this newly proposed constant.

In this work, we consider metrics on Euclidean space with nonnegative scalar curvature and rapid decay at infinity. We show that, in dimensions four and higher, any such metric is necessarily flat if its decay rate exceeds that of the Schwarzschild metric. This complements recent works by Mazurowski-Yao and You-Zhang, thereby establishing Gromov's conjecture on the rigidity of the positive mass theorem under fast metric decay in all dimensions. Our method also extend naturally to weakly asymptotically flat manifolds.

We propose a meshless Lagrangian dynamical method for approximating Wasserstein gradient flows (WGFs). The evolving measure is represented as the pushforward of the initial measure $\mu_0$ through a transport map in the weighted Hilbert space $L^2_{\mu_0}$. We approximate this map in time-dependent linear subspaces of $L^2_{\mu_0}$, whose orthonormal frames are evolved by a Dirac–Frenkel dynamical principle on a Stiefel manifold constrained to a finite-dimensional background space, adaptively constructed via local approximations of the WGF velocity field. We prove that the resulting transport map induces an absolutely continuous curve of probability measures in Wasserstein space, whose velocity is obtained by projecting the exact WGF velocity onto the background space, and we show that the approximation preserves the energy dissipation structure up to the projection error of the velocity. Moreover, for geodesically convex energies, we derive an a posteriori estimate controlling such projection error through the adaptive construction of the background space, yielding as well a bound on the approximation error of the pushforward measure in the Wasserstein metric. Numerical experiments on linear and nonlinear Fokker–Planck equations, porous-medium diffusion, and interaction energies demonstrate the accuracy of the method, its energy-dissipation properties, and the advantages of the adaptive construction.

Classical rate-distortion (RD) theory has long established the fundamental limits of lossy compression by quantifying the minimum number of bits required to represent a source under a prescribed distortion constraint. However, widely used distortion measures such as mean-squared error often fail to capture perceptual quality or semantic validity, which are increasingly central in modern learning-driven applications. Rate-distortion-perception (RDP) theory extends the RD framework by introducing perception as a third fundamental axis, quantified via distributional similarity between the source and reconstructed signals, leading to the rate-distortion-perception function (RDPF). This tutorial provides a structured overview of the coding principles underlying perception-aware lossy compression and surveys recent achievability results under different randomness assumptions. It then presents a unifying optimization viewpoint for computing the RDPF as defined by Blau and Michaeli, for both discrete and continuous sources under broad families of perceptual constraints, including f-divergences, alpha-divergences, and Wasserstein-based metrics. Special attention is given to computational tools such as alternating minimization schemes, Newton-based methods, and convex optimization formulations, as well as to analytically tractable cases such as Gaussian sources and the perfect-realism regime. Unlike recent broad surveys that emphasize generative architectures and AI-empowered communication systems, this tutorial focuses on the coding-theoretic and computational machinery needed to characterize, compute, and interpret the RDP limits. Finally, the tutorial outlines promising research directions at the intersection of information theory, neural compression, robust source coding, and perception-aware networked control systems.

For a division algebra $D$, let $K_1(D) = D^*/[D^*, D^*]$ and let $\operatorname{TK}_1(D)$ be the torsion subgroup of the abelian group $K_1(D)$. We study this torsion group for graded and valued division algebras, in parallel with the known theory of $\operatorname{SK}_1$. For a graded division algebra $E$ finite-dimensional over its center, we give exact sequences describing $\operatorname{TK}_1(E)$ in terms of $E_0$, the grade group~$\Gamma_E$, and the conjugation action of $E^*$ on $E_0$. These yield explicit formulas for $\operatorname{TK}_1(E)$ in the unramified, totally ramified, and semiramified cases. For a tame valued division algebra $D$ over its Henselian-valued center $K$, we identify the obstruction group $\mathbf H$ to a congruence theorem for $\TK(D)$. We show that if the residue field~$\overline K$ of the valuation on $K$ has characteristic $p > 0$, then $\mathbf H \cong\mu_K[p]$, the $p$-primary component of the group $\mu_K$ of roots of unity in $K$; but if $\operatorname{char}(\overline K)=0$, then $\mathbf H=1$. We further prove a short exact sequence $$ 1\,\longrightarrow \,\mathbf H\, \longrightarrow \,\operatorname{TK}_1(D)\, \longrightarrow\, \operatorname{TK}_1(\gr(D))\, \longrightarrow \,1, $$ where $\gr(D)$ is the associated graded division algebra determined by the valuation on $D$ obtained from the valuation on $K$. We also prove a stability theorem for a graded division algebra $E$ with quotient division ring~$q(E)$, i.e., $$ \operatorname{TK}_1(E)\,\cong \,\operatorname{TK}_1(q(E)), $$ together with a new proof of the corresponding stability theorem for $\operatorname{SK}_1$. As applications, we obtain graded analogues of Motiee's primary decomposition and scalar-extension results for torsion Whitehead groups.

We study an insurance contract-design problem under moral hazard, endogenous participation, and strategic risk interdependence. Because the resulting $N$-agent game suffers from the curse of dimensionality, we approximate the strategic interactions via a heterogeneous mean-field game. We rigorously establish the existence of a lower-level mean-field Nash equilibrium using measurable selection arguments and the Kakutani fixed-point theorem. By proving the $L^1$-Lipschitz continuity of the aggregate participation threshold, we further establish equilibrium uniqueness via a contraction mapping. We then embed this mean-field response into the insurer's upper-level Stackelberg optimization problem. We formulate the objective through general performance envelopes to accommodate potential equilibrium multiplicity, proving the existence of upper-level $\varepsilon$-optimal contracts, and demonstrating the existence of an exact Stackelberg equilibrium under the uniqueness regime. We conclude by extending the model to finite contract menus, providing numerical evidence that multi-contract screening improves the principal's expected payoff in interdependent risk environments.