Papers

Simulating dispersive shock waves (DSWs) using physics-informed neural networks (PINNs) presents significant numerical difficulties. In particular, the square-root initial condition exhibits singularity near the origin, making it difficult to fit accurately. This initial fitting error is amplified through nonlinear evolution and gradually accumulates during extrapolation, imposing high demands on the long-term extrapolation capability of numerical methods. Accordingly, this paper proposes an improved physics-guided multistage neural network (IPgMSNN) framework. IPgMSNN is built upon the original PgMSNN framework, with its core innovation focused on the third-stage training strategy. Specifically, online fine-tuning and a weighted target mechanism are introduced, which effectively suppress error accumulation during long-term extrapolation. Numerical experiments systematically evaluate IPgMSNN from three perspectives: forward problem solving, model stability, and parameter inversion. Experimental results demonstrate that IPgMSNN effectively captures high-frequency wavefront details, maintains long-term extrapolation stability, and achieves high-precision parameter inversion, providing an efficient solution for deep learning-based simulation of dispersive nonlinear systems.

We show that the shifted log-sine integrals with odd entries, introduced by Umezawa, are multiple zeta values. In particular, they define an algebra homomorphism from the f-alphabet algebra to the algebra of multiple zeta values. The proof uses a motivic realization at sixth roots of unity and the Galois descent from level six to level one studied by Glanois. Finally, we give an interpretation of our results using Zinbiel algebras and introduce iterated cotangent integrals.

We implement Ozaki Scheme II (residue number system + Chinese remainder theorem), which reduces multiple-precision dense matrix multiplication to a sequence of low-precision, high-throughput integer or floating-point GEMMs on CPUs. Two backends are built on top of a shared CRT reconstruction stage: (a) exact INT8 x INT8 -> INT32 tile products on Intel AMX, and (b) binary64 DGEMM, the CPU construction of the original Ozaki Scheme II paper. On a two-socket Xeon Gold 6526Y (Emerald Rapids, 32 cores), we evaluate significand precisions of 53-2048 bits and matrix dimensions N = 256-8192. The results are always within 1 ulp of a high-precision MPFR reference (essentially correctly rounded), while running up to 167x faster than a naive MPFR matrix product, up to 588x faster than BNCmatmul's Strassen multiplication, and 9-78x faster than Ozaki Scheme I (FP64 slicing + OpenBLAS DGEMM). The break-even point between the two backends is approximately N = 2048: below it, the binary64 backend wins thanks to its smaller number of moduli; above it, the AMX-INT8 backend wins as the GEMMs dominate. We further port the implementation to AArch64 (NVIDIA GB10: Cortex-X925 x 10 + Cortex-A725 x 10). Since this machine lacks SME/SME2, the INT8 kernel uses the SMMLA matrix-product instruction of the SVE2 i8mm extension. We obtain an exact INT8 GEMM sustaining 6.5 TOPS and, still within 1 ulp across all conditions, speedups of 14-89x over BNCmatmul's Ozaki Scheme I (OpenBLAS-linked routine) and 6-19x over a fairness-adjusted OzI-best variant. The paper also includes a tutorial introduction to Ozaki Scheme II (Section "Introduction to Ozaki Scheme II") and a quantitative explanation of why this seemingly GPU-specific technique is fast on CPUs as well.

We determine exactly the range of the modular Hamiltonian of an interval in a chiral conformal field theory over states of bounded energy. For an interval of length $R$ in a theory of central charge $c$, and states of mean energy at most $E$, the supremum of the modular energy is $M_{\max} = \frac{\pi R}{2}E + \frac{c}{6} - \frac{\pi c^{2}}{288\,RE} + O((RE)^{-2})$, and neither constant can be improved. The infimum is logarithmic, $M_{\min} \simeq -\frac{c}{12}[\ln(24\pi RE/c)-1]$, so the range is linear above and logarithmic below, the two meeting as $RE \to 0$; it is strictly wider above, by more than $\frac{2}{5}\pi RE$ at every energy, which reduces after a Stieltjes substitution to the positivity of a variance. Both bounds are attained, the one-parameter family of weights proving them being the Legendre structure of the answer. The two ingredients separate cleanly. The coefficient $\pi R/2$ is Möbius representation theory alone: the complementary weight is a translate of the special conformal generator, hence positive, so for the full modular Hamiltonian the bound needs no additive constant. The constant $c/6$ is precisely the price of truncating that weight at the endpoints, supplied by the quantum energy inequality of Fewster and Hollands; the saturating state places its negative energy exactly there. The constant is accordingly additive over entangling points, being $nc/6$ for $n$ disjoint intervals of equal length. Composing with positivity of relative entropy gives a sharp Bekenstein bound, $\Delta S_I \leq \frac{\pi R}{2}E + \frac{c}{6}$, recovering the original form of that inequality from the modular form that by itself imposes no constraint.

Karasev proved, in dimensions two and three, a sharp surface-area counterpart of a polyhedral extension of Vaaler's theorem [8, Theorem 1.2]. We remove the dimension restriction. More precisely, if an $n$-dimensional convex polytope $P$ contains the origin in its interior and the affine hull of every nonempty proper face of codimension $k\in\{1,\ldots,n\}$ is at distance at least $\sqrt{k}$ from the origin, then $$ \mathcal{H}^{n-1}(\partial P)\geqslant n2^n. $$ Consequently, the boundary of every $n$-dimensional linear section of the cube $[-1,1]^N$ has $(n-1)$-dimensional measure at least $n2^n$. This confirms, in all dimensions, a conjecture of Grigory M. Ivanov recorded by Karasev [8, Section 1]. The proof combines the Rogers–Karasev flag decomposition with a dimension-free Gaussian comparison for orthoschemes. Its main step is an ordered-square substitution which converts all face-distance assumptions into a pointwise domination of a single positive integral. We also disprove the naive extension to all skeletal measures, formulate a ridge-skeleton conjecture, and establish two partial results toward it.

The classical Veronese formula of Goto–Watanabe identifies the canonical module of a Veronese subalgebra with the corresponding Veronese module. We prove the analogous compatibility for graded duals under a module-finite extension whose upper ring is equidimensional, satisfies Serre's condition $(S_2)$, and has dimension at least two, and obtain the formula for the $b$-invariant. This yields explicit two-sided bounds for canonical traces: the lower bound is controlled by the Loewy length of the canonical-trace quotient, whereas the upper bound is governed by the $a$- and $b$-invariants, with an $S_2$-ification refinement for unmixed rings. We also characterize when all sufficiently large Veronese subalgebras of standard graded level algebras are nearly Gorenstein when the $a$-invariant is negative. Applications include Segre products, Stanley–Reisner rings and determinantal rings.

In safe hypothesis testing with test supermartingals, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $\alpha\in(0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $\frac{1}{\alpha}$. Due to an inherent asymmetry, the type-II error does not have such guarantees: a heavy concentration of the probability on the lower tail of the log-increments can lead to one catastrophic bet that undoes any amount of accumulated evidence. This paper studies how different hypotheses on those lower-tail probabilities lead to different bounds on the type-II error of the sequential test. They all reduce to one master inequality, which bounds the type-II error at level $\alpha$, at a fixed horizon and sequentially, in terms of a one-sided Legendre transform of the (inverse-)moment generating function of the e-variables, evaluated at one number: the amount by which the lower bound of the accumulated e-powers exceeds $\log\frac{1}{\alpha}$. And, the step is lossless, in the sense, that it extracts exactly a constrained information projection. Every bound presented here is a corollary, obtained by a certain majorant of the above function. The hypotheses are: a finite negative moment; an exponentially small crash probability with a moment on the winning side; a wealth floor with a conditional variance, and its Bernstein variant, which interpolates between a Gaussian regime set by the variance and an exponential one set by the scale; a sub-Gaussian or bounded-tilt lower tail; bounded log-increments; and i.i.d. increments, where the majorant is the truth. We also provide an empirical-Bernstein variant. Each hypothesis may either be read as a condition on the e-variables one has, or as the price of betting with an approximation to the likelihood ratio rather than the ratio itself, which satisfies the weakest condition for free.

In safe hypothesis testing with test supermartingales, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $\alpha\in(0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $1/\alpha$. Due to an inherent asymmetry, the type-II error behaves differently. We prove two things about the latter, for a simple null and alternative. First, the mean growth rate $\mathbb{E}_{P_1}[\log E]$, the e-power, that Kelly betting and growth-rate-optimal e-variables maximise, bounds nothing on its own. For every level $c>0$, every $\alpha$ and horizon $t$ we construct e-variables of conditional e-power exactly $c$ whose probability of not rejecting by $t$ is arbitrarily close to one. It forces eventual rejection, but no finite-horizon guarantee follows. Second, the quantity that does control the type-II error is the Chernoff-Stein exponent of an e-variable, $\Lambda(E)=\sup_{s\ge0}\{-\log \mathbb{E}_{P_1}[E^{-s}]\}$, whose range is exactly determined: $\sup_E \Lambda(E)=\mathrm{KL}(P_0\|P_1)$, the classical Chernoff-Stein exponent, and so the ceiling of its own per-e-variable form. One conditional application of Hoelder's inequality per step gives it, for every test supermartingale on an arbitrary filtered space, with no independence or product structure; the i.i.d. case adds that it is matched, and attained by nothing. The e-power has its own ceiling, $\mathrm{KL}(P_1\|P_0)$, and that one is attained, $P_0$-a.s. uniquely, by the likelihood ratio $R$. The two optima are the same divergence in opposite arguments, at opposite ends of the flattened family $R^{\beta}/\mathbb{E}_{P_0}[R^{\beta}]$: the ceiling as $\beta\downarrow0$, $R$ at $\beta=1$. Which $\beta$ is best is settled by the horizon, exactly: $R$ is optimal at $t=\log(1/\alpha)/\mathrm{KL}(P_1\|P_0)$ alone, beaten by sharpening $(\beta>1)$ below it and by flattening above.

Let $r\ge 3$ and $k\ge 2r+1$ be fixed integers. We determine, for all sufficiently large $n$, the maximum adjacency-tensor spectral radius of an $n$-vertex $r$-uniform hypergraph containing no Berge cycle of length at least $k$. Write $s=\left\lfloor\frac{k-1}{2}\right\rfloor$. If $k=2s+1$ is odd, the unique extremal hypergraph consists of all $r$-sets containing at most one vertex outside a fixed $s$-set. If $k=2s+2$ is even, one additionally includes all $r$-sets containing a fixed pair outside the $s$-set together with $r-2$ vertices inside it. Consequently, the maximum spectral radius is given as \[ \operatorname{spex}_r(n,k) = \left[ \binom{s}{r-1} \left(\frac{r-1}{s}\right)^{(r-1)/r} +o(1) \right]n^{(r-1)/r}. \]

We prove the Tate conjecture for abelian fourfolds over finite fields. This is the first resolution of the conjecture for all abelian varieties of a fixed dimension over finite fields since the work of Tate in the 1960s. The proof relies on techniques from Ancona's proof of the standard conjecture of Hodge type for abelian fourfolds, and ultimately reduces to Markman's results on the algebraicity of Weil classes on complex abelian varieties. Combining the above case of the Tate conjecture with theorems of Ancona and Kahn, we deduce that the standard conjecture on homological versus numerical equivalence holds for abelian fourfolds over arbitrary fields. This completes the proof of the standard conjectures for abelian fourfolds.

A quantum $(r,\rho)$-locally recoverable code ($(r,\rho)$-qLRC) is a quantum code in which every qudit can be recovered from at most $r+\rho-1$ other qudits, even after $\rho-1$ additional erasures inside the recovery set. The bounds currently known for this class, namely the Singleton-like and the GG Singleton-like bounds, are alphabet independent and are therefore loose for small-to-moderate qudit dimensions. In this letter, we derive three alphabet-dependent upper bounds for pure $(r,\rho)$-qLRCs obtained through the Hermitian CSS construction: a Griesmer-like, a Plotkin-like, and a sphere-packing-like bound. We further establish the asymptotic hierarchy among these bounds and identify the relative-distance regions in which each of them yields the tightest rate constraint.

Let $G$ be a finite connected simple graph, let $uv$ be a nonedge, and let $s$ be a starting vertex. We study the exact change in the expected cover time of simple random walk when $uv$ is inserted. A killed Green matrix update, combined with target-set inclusion-exclusion, gives an exact formula using only the original graph. The response can have either sign. Our main result is a nonlocality theorem. For every radius $r\geq 1$, we construct two marked configurations whose ambient-degree-labelled radius-$r$ neighbourhoods at $s,u,v$ are isomorphic but whose fixed-start cover-time responses have opposite signs. The construction also matches the three marked degrees, the marked distance, and the effective resistance $R_{uv}$. The two graphs have different orders; equal-order nonlocality remains open. We complement this result with a conductance interpolation theorem and an occupation interpretation of its high-conductance coefficient. After contracting $u$ and $v$, that coefficient is a positive multiple of the expected pre-cover occupation of the contracted vertex, and its zero case is classified exactly. A path with two pendant insertion endpoints is solved for all starting vertices and shows a sharp change of sign across the start set.

We investigate the action of the affine Yangian $Y_{t_1,t_2}(\widehat{\mathfrak{gl}}_1)$ on the singular cohomology of the moduli space of stable sheaves $\mathcal{M}$ on a smooth projective surface $S$. Through an intersection-theoretic analysis of nested moduli spaces, we obtain a proof of the representation $$Y_{t_1,t_2}(\widehat{\mathfrak{gl}}_1) \curvearrowright H_{\mathcal{M}}.$$ This intersection-theoretic approach extends classical constructions to the full Yangian structure and circumvents the $K$-theoretic machinery traditionally required to establish this action.

Pro-nilsystems arise naturally in ergodic theory through the study of multiple ergodic averages. The Host-Kra structure theorem implies that every factor of an ergodic $s$-step pro-nilsystem is again an $s$-step pro-nilsystem. No proof of this fact independent of the structure theorem was previously known. In this thesis, we give such a proof, using only arguments intrinsic to nilsystems. The main new ingredient is a local rigidity result for self-joinings of nilsystems: every ergodic self-joining sufficiently close to the diagonal joining is necessarily the graph joining of an automorphism. We prove this geometrically by showing that nilmanifolds cannot contain arbitrarily small nontrivial subnilmanifolds. We also obtain a direct proof of the analogous factor-closure result for transitive topological pro-nilsystems, previously known as a consequence of the Host-Kra-Maass topological structure theorem. Finally, we give a detailed account of Tao's deduction of a weak structure theorem directly from the Green-Tao-Ziegler inverse theorem for the Gowers norms. Combined with the factor-closure of pro-nilsystems, this yields a new proof of the Host-Kra structure theorem. The principal new results of this thesis also appear in a joint paper with Asgar Jamneshan (arXiv:2604.02089). This thesis provides a more self-contained treatment, including the necessary background on dynamical systems and nilsystems, as well as full details of Tao's argument.

Decisions and their consequences can be described and analyzed by means of decision trees. The decisions themselves depend on questions that, whenever possible, should be answered with yes or no. The original BSS machines over the real numbers are graphs with computation nodes and branching nodes for decisions. The evaluation of BSS machines and algebraic decision trees in computer-aided geometry which have been introduced for various types of numbers generally involves the evaluation of systems of literals of first-order logic. Flowchart-like representations and decision trees are also helpful for analyzing decisions made by BSS RAMs over first-order structures. All algorithms defined by machine-oriented programs of BSS RAMs can be illustrated using flowcharts, walks, and paths in program trees. The basic structures of these tools are graphs and their visual representations can help to characterize the behavior of individual BSS RAMs and other first-order machines. We offer a theoretical framework for linking various models. Here, we introduce program paths and transition systems for transforming partial configurations which are defined syntactically and can be extended and refined later. Finally, we define standard orders for program paths and present algorithms for enumerating finite program paths.

Even for tension-dominated fracture propagation, porous materials may deform plastically adjacent to the propagating fracture. As is common for porous materials, existing phase-field models typically employ a non-associative flow rule for plasticity, and a Helmholtz free energy based on strain and fluid pressure. This work revisits the hydromechanically coupled formulation of the phase-field model for fracture in poro-elastoplastic media by analyzing the strength surface and fracture driving force. Our analyses show that these common choices of flow rule and free energy will lead to a discontinuous strength surface across the tension-compression transition. A non-associative flow rule introduces a jump at the strength surface, while treating fluid pressure-rather than fluid content-as the independent variable in the Helmholtz free energy omits a coupling term from the phase-field driving force, also breaking continuity. Incorporating an associative Drucker-Prager flow rule and this omitted coupling term ensures a continuous strength surface and the accurate fracture driving force. The proposed model exhibits improved accuracy in hydromechanical responses when compared against the analytical solution of the Kristianovich-Geertsma-de Klerk hydraulic fracturing benchmark. Numerical simulations of hydraulic fracturing and biaxial compression in poro-elastoplastic media show that the model can reproduce both shear-dominated fractures induced by mechanical disturbance and tension-dominated fractures driven by fluid injection in saturated porous media.

For any $m, s \in \mathbb{N}$, we study the number $N_{m\times s,q}(\kappa, \beta)$ of solutions $(x_1,\ldots,x_s) \in (\mathbb{F}_q)^s$ of the monic system of diagonal equations $$ X_{1}^{k_i} + \cdots + X_{s}^{k_i}= \beta_i, \qquad (1\le i \le m), $$ with $\kappa=(k_1,\ldots,k_m) \in \mathbb{N}^m$ and $\beta=(\beta_1,\ldots,\beta_m) \in (\mathbb{F}_q)^m$. We show that this number can be obtained in terms of some data of diagonal GP-graphs $\Gamma(\kappa,q)$. This is a new family of graphs that we introduce here, $\Gamma(\kappa,q)$, with $\kappa = (k_1,\ldots,k_m) \in \mathbb{N}^{m}$, is the directed graph with vertex set the finite field $\mathbb{F}_q$ and there is an arc from $u$ to $v$ if and only if $v-u \in R_{\kappa} = \{ (x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_{q}^*\}$. In particular, we give three different expressions for $N_{m\times s,q}(\kappa, \beta)$: one in terms of walks, another in terms of adjacency matrices of $\Gamma(\kappa,q)$ and the last one in terms of the spectrum of $\Gamma(\kappa,q)$. Finally, we explicitly derive combinatorial formulas for the number of solutions $N_{m}(s,q) = N_{m\times s,q}(\kappa_\ell, 0)$ of monic homogeneous systems of diagonal equations of the form $$ X_1^{q^{\ell_i}+1} + \cdots + X_s^{q^{\ell_i}+1} = 0 \qquad (1\le i \le m),$$ with $\kappa_\ell=(\ell_1,\ldots,\ell_m)=(1,3,\ldots,2m-1)$ and $m\ge 2$, via the known spectrum of Hermitian-form graphs, which can be viewed as diagonal GP-graphs. For any $m,s \in \mathbb{N}$, we give general summation and recursive formulas for $N_m(s,q) \in \mathbb{Z}[q]$. For the small cases $N_{1}(s,q)$, $N_{2}(s,q)$ and $N_{m}(s,q)$, with $1\le s \le 5$, we give explicit expressions.

It is known that generalized hyperbolicity implies the shadowing property for invertible bounded linear operators on a Banach space. Whether the converse holds has been a central open problem in linear dynamics and has been conjectured to have a positive answer. We show that this conjecture fails on general Banach spaces by constructing a counterexample, whereas it holds on separable Hilbert spaces. The distinction is explained by the gap that may occur on Banach spaces between surjectivity and right invertibility, a gap that disappears on Hilbert spaces. The main ingredients of the proofs are a recent spectral characterization of shadowing in terms of the surjective spectrum and a new characterization of generalized hyperbolicity in terms of right resolvent functions near the unit circle.

We discuss the advantages of the Calculus of Moving Surfaces over the Euler-Lagrange equation for optimization problems originating in Geometry. An extension of Tensor Calculus, it provides tools for analyzing geometric quantities directly rather than their coordinate representations. This allows us to avoid the many difficulties associated with the use of coordinates, from the untenable complexity of analytical expressions to the virtual impossibility of recovering the geometric interpretation of the final result. As an illustration, we analyze the brachistochrone and give its geometric characterization in terms of its curvature.

Most modern optimizers form their momentum as an exponential moving average (EMA) of past gradients, forgetting every direction at one fixed rate. However, the inputs a deep network sees during training can be highly anisotropic, with a few directions queried frequently while most are seen rarely. Preconditioning methods address this anisotropy by wrapping extra processing around this buffer and leave the momentum update itself unchanged. We propose Activation-Keyed Momentum (AK-Momentum), which builds direction-awareness into the momentum update rule. The gradient of a linear layer splits into an input activation that acts as a $\textit{key}$ and an output-side error that acts as a $\textit{value}$. Keying on that activation, AK-Momentum updates the momentum buffer by the canonical delta rule, so each direction is forgotten at a rate set by how often it appears. We prove that it is a valid momentum, that it applies the input-side curvature correction without matrix inversion, and that it clears stale directions faster than EMA under both a fixed and a drifting optimum. It is a drop-in replacement for the momentum buffer of any optimizer, its coefficient transfers across widths under $\mu$P, and its extra compute stays between $22.2\%$ and $25.0\%$ of a gated-MLP block's linear cost with no persistent memory. In FineWeb-Edu pretraining, AdamW with AK-Momentum (AK-AdamW) reaches AdamW's validation loss in up to $46.64 \pm 4.52\%$ fewer steps at 67M and $22.14 \pm 0.81\%$ at 370M over three seeds, and the gain persists at 1B on a Chinchilla-optimal budget. A Muon baseline tuned under the same protocol sits above AK-AdamW at both language-model scales, and the gain holds for SGD, ResNet-18, and ViT-Tiny on CIFAR-10. Training-time diagnostics confirm the predicted mechanism, better gradient tracking and healthier input directions. Code is available at https://github.com/euijinh/AK-Momentum.

We establish a general framework for developing fast sampling and counting algorithms for stoquastic spin systems at high temperature. Our framework is based on a rapidly mixing Markov chain for polymer models and a subcritical percolation process for sampling individual polymers. We apply our framework to obtain fast algorithms for approximating the partition function and sampling from the thermal distribution of (1) general stoquastic spin systems, (2) ferromagnetic Heisenberg models, and (3) antiferromagnetic Heisenberg models on bipartite graphs. For the Heisenberg models, we obtain an improved bound on the inverse temperature by using their respective cycle and loop representations.

Quadratic manifold-based model order reduction offers a viable pathway to circumvent the limitations of linear subspaces for linear control systems characterized by slow Kolmogorov $n$-width decay. However, a system-theoretic framework for constructing such quadratic approximations remains absent from the literature. This paper presents a system-agnostic, optimization-free framework for the direct construction of quadratic projection matrices. We prove that the synthesized reduced-order model matches the nonlinear moments of the full-order system and preserves its exact center manifold mapping, thereby ensuring asymptotic tracking of steady-state outputs under specific input classes. Numerical results on transport-dominated benchmark problems, namely, the one-dimensional damped wave and advection equations, show that the proposed framework achieves high-fidelity trajectory reconstruction within a significantly reduced-dimensional state space, yielding substantial online computational savings.

A shadow (or projection) is obtained from a link diagram by ignoring the over/under information at each crossing. Given a fixed link $L$ we investigate the complexity of deciding whether an input shadow $S$ can be resolved into $L$, that is, whether we can assign over/under information to its crossings to obtain a diagram of a link isotopic to $L$. We show that if $L\in\{3_1,4_1,5_1,5_2,6_2,L2a1, L4a1, L5a1, L6n1\}$ then there exists a linear-time algorithm that decides whether an input shadow $S$ resolves into $L$.

We study the dynamics of the kinetic wave equation associated to the three dimensional Schrödinger equation close to Rayleigh-Jeans equilibria. We first prove that the linearised operator generates a semigroup of contractions in $L^2((0,\infty);\sqrt \omega \dd\omega )$. Considering the family of nonsingular Rayleigh-Jeans spectra, we prove that the linearised operator possesses a spectral gap, despite the non-compactness of the integral collisional operator, and thus obtain an exponential relaxation for the linear semigroup. We then prove bilinear and trilinear estimates in the relevant norm for the nonlinear terms and deduce global well-posedness and exponential relaxation for sufficiently small relative perturbations of Rayleigh-Jeans equilibria. To our knowledge, this is the first global strong well-posedness and asymptotic stability result near a nonzero thermodynamic equilibrium for the full spatially homogeneous four-wave kinetic equation associated with the cubic nonlinear Schrödinger equation.

Universally (or geometrically) multiplicative norms on Banach algebras over complete non-archimedean fields were used by Berkovich in his works on non-archimedean geometry, and later they were studied in some detail by Poineau. In this paper, we perform a more thorough study of the question when multiplicativity, spectrality and spectral multiplicativity of norms on algebras over real valued fields are preserved by ground field extensions. We obtain precise criteria quite analogous to the classical theory of geometric irreducibility and reducedness. In particular, we generalize the results of Poineau in a few aspects.

We study a one-dimensional free boundary problem arising in combustion theory, where the motion of the interface is governed by a prescribed Neumann boundary flux and a zero Dirichlet boundary condition. We treat both the half-line case and the bounded interval case. For both settings, we employ maximal $L^p$-$L^q$ regularity as our main analytical tool. In the half-line case, the solutions need not decay at infinity, even though the spatial derivatives belong to $L^q(\mathbb{R}_+)$. To handle the evolution law of the free boundary, we introduce a derivative formulation that avoids second-order boundary traces. By combining maximal $L^p$-$L^q$ regularity and Schauder estimates, we establish the local-in-time existence, uniqueness, and regularity of solutions, as well as the evolution law of the free boundary.

We prove that a finite group $G$ is solvable whenever $MN=G$ for every pair of nonconjugate maximal subgroups $M,N<G$. Equivalently, every finite nonsolvable group has two nonconjugate maximal subgroups whose setwise product is proper. This gives a negative answer to Problem 10.34 of the Kourovka Notebook. As an application, we also answer an open question raised by Guo.

We study one-dimensional screening when the Spence-Mirrlees condition fails along a monotone dividing curve, on which all types value marginal output equally. A single ratio, marginal distortion over marginal rent, governs the problem. When the relaxed solution stays in the positive region, non-decreasing optima in the positive single-crossing region are continuous if the ratio increases in the decision. When it crosses a flat curve and marginal rents are separable, every continuous contract is improved by a jump. When it crosses a strictly monotone curve, a contract that pools and jumps continues along the mirror of the pool if the ratio does not increase along it. Optimality among all implementable allocations, deterministic or random, is proved by a measure on the binding constraints whose weights are the same ratio; with constant curvatures and no crossing, it is also necessary. As applications, we solve problems of procurement and nonlinear pricing.

High Bandwidth Memory (HBM) systems can exhibit partial service rather than only full service or complete isolation: a controller-visible service unit may deliver full, reduced, or zero bandwidth because of sub-channel isolation, lane remapping, or protection overhead. This paper develops an exact multistate reliability framework in which each service unit carries an arbitrary finite set of bandwidth states and reliability is the probability that aggregate delivered bandwidth meets a demand. The binary k-out-of-n model is recovered as a special case, while closed-form binary-mapping error relations quantify mean-bandwidth distortion and provide a screening test for the simpler abstraction. For exact single-threshold evaluation a threshold-pruned multistate Binary-Addition-Tree (TP-mBAT) algorithm is proposed. It is deliberately regime-specific: fixed-grid dynamic programming is preferable on a compact common grid, where a 16-unit commensurate control required 273 pruned-DP updates versus 362,506 TP-mBAT node visits. On a reproducible 14-unit incommensurate benchmark, TP-mBAT is compared against a dynamic program carrying the same threshold rules, so that no baseline is weakened. Both expand the same state space, 6,862 nodes against 6,861 updates, and the separation lies entirely in retained state: 17 traversal entries against 412,121 probability states at central demand, reducing measured peak storage from 25.02 MB to 1,152 B. An exact probability-transfer sensitivity identifies when moving mass from a degraded state to a higher-bandwidth state changes system success, and a reserved-unit floor model admits a third exact pruning rule that is vacuous without such floors. A latent package-state mixture captures shared stress, where ignoring dependence overstates reliability by 8.73 percentage points.

We show that quasitoric manifolds can be equipped with an invariant metric for which the torus action is polar. More generally, we show that an infinitesimally polar action of a torus admits an invariant Riemannian metric making it polar if and only if the real orbifold Euler class of its quotient vanishes. Because the orbit spaces of quasitoric manifolds are contractible simple polytopes, this cohomological obstruction trivially vanishes.