Papers

This article concerns Wiener amalgam spaces % , recalls their basic properties and provides some hints about their usefulness in various branches of Harmonic Analysis. Despite the fact that the underlying construction principles % of Wiener amalgam spaces is are quite easy to understand and basic facts follow naturally by simple rules, these spaces have not obtained the same popularity as certain other function spaces which are much more complicated to describe and often just serve a very particular purpose. \newline \indent This situation has motivated the author to provide here a summary of the foundations of the theory of Wiener amalgam spaces (and the motivation behind their construction) and a selection of relevant applications, some 45 years years after the key paper published in 1983. \newline \indent We recall first that the so-called classical Wiener amalgam spaces using local $\HFLpsp$-norms combined with a global $\HFlqsp$-behaviour are already quite useful, e.g.\ for an improvement of the Hausdorff-Young Theorem with some interesting consequences for Sobolev algebras. However, the main emphasis will be based on the idea of allowing more general local components (describing for example smoothness or membership in the Fourier algebra). This opened the door to the introduction of modulation spaces, which are now recognized as standard tools in time-frequency analysis. \newline \indent We will demonstrate in this article how Wiener amalgam spaces methods can be used to prove the Sobolev embedding theorem or determine the pointwise multipliers of Sobolev algebras. We also demonstrate that the space of multipliers from the classical Wiener algebra $\HFWCOliRd$ into its dual can be identified with $\HFSOPRd$, the space of mild distributions. }

For an integer $q\ge 2$ and a graph $F$ satisfying $q\mid e(F)$, the zero-sum Ramsey number $R(F,\mathbb Z_q)$ is the least integer $n$ such that every edge-labeling $w\colon E(K_n)\to \mathbb Z_q$ contains a copy of $F$ whose edge-label sum is zero in $\mathbb Z_q$. Write $K_{s,t}$ for the complete bipartite graph with $s$ vertices on one side and $t$ vertices on the other side. We prove that for every $q\ge2$, there is an explicit threshold $S(q)$ such that $R(K_{s,qk},\mathbb Z_q)=s+qk$ for all $s\ge S(q)$ and all $k\ge1$. We also determine the zero-sum Ramsey number of $K_{s,3k}$ over $\mathbb Z_3$ for all $s\ge2$ and $k\ge1$. We prove that $R(K_{s,3k},\mathbb Z_3)=s+3k$, except when $s=2$ and $k\ge1$, or when $s\in\{3,4,5,7\}$ and $k=1$. In these exceptional cases, $R(K_{s,3k},\mathbb Z_3)=s+3k+1$. In particular, this shows that the threshold $S(q)$ is best possible for \(q=3\).

We describe an elementary sequential realization of the Banach Gelfand triple (S0(R^d), L2(R^d), S0'(R^d)). Here S0(R^d) is a Segal algebra of test functions, L2(R^d) is the usual Hilbert space, and S0'(R^d) is its dual space of mild distributions. This framework is fundamental for Gabor analysis and provides a natural setting for the generalized Fourier transform and the short-time Fourier transform. Inspired by Lighthill's sequential approach to tempered distributions, we construct an extended domain for the short-time Fourier transform from equivalence classes of extended mild Cauchy sequences, abbreviated as ECmiCS. Their representatives are sequences of bounded continuous functions. The construction avoids Lebesgue integration and the theory of tempered distributions. Our main result identifies the resulting sequential space canonically with S0'(R^d), thereby recovering the Banach Gelfand triple in an elementary form.

In this review, we offer a comprehensive survey of emerging techniques in gradient-based optimization, with a particular emphasis on the interplay between ordinary differential equation (ODE) perspectives and their extensions into discrete Lyapunov analysis. We begin by examining the acceleration mechanisms underlying Nesterov's accelerated gradient method for strongly convex functions (NAG-SC) and Polyak's heavy-ball method, identifying the gradient-correction term as the primary driver of acceleration. This mechanistic insight is substantiated through high-resolution ODE modeling and the systematic construction of Lyapunov functions. We then synthesize recent advancements in convex optimization regarding NAG and its proximal generalization, the fast iterative shrinkage-thresholding algorithm (FISTA). Key topics include the accelerated convergence of gradient norms, underdamped acceleration, linear convergence under strong convexity, and novel Lyapunov frameworks for establishing convergence and monotonicity properties of generalized accelerated methods. Furthermore, we demonstrate how these ODE approximations and Lyapunov techniques can be extended to provide a unified framework for analyzing advanced optimization algorithms, including the alternating direction method of multipliers (ADMM), the primal-dual hybrid gradient (PDHG) method, and their respective accelerated variants. Finally, we discuss recent progress in minimax optimization and outline future directions for extending Lyapunov-based analysis to saddle-point problems.

We study the Bayesian fixed-budget best-arm identification problem in which a learner can abstain from making a terminal recommendation. Subject to an abstention budget $\alpha$, we analyze the probability of undetected error–the risk of recommending a suboptimal arm without abstaining. Our central finding is that abstention induces a phase transition: without abstention, the error probability decays polynomially in the sampling budget $T$; in contrast, introducing any small positive abstention budget shifts this to an exponential decay. For Gaussian priors and rewards, in the regime $T\to\infty$ followed by $\alpha\downarrow0$, we establish exact matching information-theoretic lower bounds and algorithmic upper bounds on the optimal error exponent, which takes the form $\exp(-\frac{\alpha^{2}T}{8\kappa_{\nu}^{2}})$. The hardness parameter $\kappa_{\nu}$ represents the prior density of the top-two gap at zero, highlighting that nearly tied instances drive the fundamental error. We introduce an adaptive algorithm, PGWS, that successfully achieves this optimal exponent by expending its abstention budget on statistically ambiguous instances. We further demonstrate that this polynomial-to-exponential improvement is exclusively a Bayesian phenomenon–in the frequentist setting, abstention only affects lower-order exponent terms. We also extend our results beyond the Gaussian model.

We develop an elementary approach to convolution and Fourier analysis on a locally compact Abelian group (G), based on bounded measures and bounded uniform partitions of unity. In earlier work, the author introduced convolution and the Fourier–Stieltjes transform on the Banach space (M(G)) of bounded measures, viewed as linear functionals, in a direct Euclidean setting. The present paper constructs arbitrarily fine bounded uniform partitions of unity on general locally compact Abelian groups. The construction is designed to avoid structure theory and does not presuppose Haar measure or Lebesgue integration. It is then used to establish a natural convolution-module structure of (M(G)) on a broad class of homogeneous Banach spaces on (G). This class includes (L^p(G)), for (1\leq p<\infty), the Fourier–Stieltjes algebra, and, in particular, Segal algebras. After introducing Haar measure, we identify (L^1(G)) with the closure in (M(G)) of the measure-embedded space (C_c(G)). We prove that the homogeneous Banach spaces under consideration are essential (L^1(G))-modules. Consequently, standard approximate identities act in the expected manner and converge strongly to the identity operator. The method follows the spirit of Hans Reiter: it avoids the customary reliance on LCA-group structure theory and on vector-valued integration arguments based on duality. It is intended as a foundation for a subsequent elementary treatment of the extended Fourier transform in the Banach Gelfand triple generated by a Segal algebra.

Given a graph $G$, the domination number $\gamma(G)$ is the minimum cardinality of a dominating set in $G$, and the packing number $\rho(G)$ is the maximum cardinality of a set of vertices that are pairwise at distance at least $3$. The ratio between these parameters has been widely studied in several graph classes. It is known that $\gamma(G) \le 2\rho(G)$ for claw-free subcubic graphs, up to finitely many exceptions, and that $\gamma(G) \le 32\rho(G)$ for unit disk graphs. In this paper, we improve the latter bound by showing that $\gamma(G) \le 16\rho(G)$ for a unit disk graph $G$. For the former bound, we show that it can be improved in the cubic bridgeless setting; more precisely, every bridgeless claw-free cubic graph $G$ satisfies $\gamma(G) \le \frac{7}{4}\rho(G) + \frac{5}{6}$. These results are not tight. In fact, we give example of an infinite family of bridgeless cubic graphs $G$ with $\gamma(G) = 5\rho(G)/4$ and an infnite family of unit disk graphs $G$ in which $\gamma(G) = 3\rho(G)$.

We propose second-order-in-time parametric finite element methods for surface diffusion of closed curves in two dimensions and closed surfaces in three dimensions. The construction is based on exact geometric variation identities along a quadratic temporal interpolation path. The induced area variation in 2D is evaluated exactly by Simpson's rule, while the induced volume variation in 3D is evaluated exactly by Boole's rule. The resulting fully discrete schemes preserve the enclosed area or volume exactly, without introducing an auxiliary Lagrange multiplier for the geometric constraint. They can be assembled on BGN-predicted auxiliary geometries and are therefore compatible with existing second-order BGN-type implementations. Numerical experiments demonstrate the expected second-order behavior, area/volume conservation, and good mesh quality for both curve and surface evolutions.

This study focuses on a mathematical model of Alzheimer's disease involving $\beta$-amyloid, cellular prion protein and their complex. The global asymptotic stability of the model indicates that the complex continues to induce neuronal damage regardless of the initial states. To investigate the dynamics of this system, we have rigorously proved that when the formation rate of new plaques is zero, the system is unconditional globally asymptotically stable without any limitation proposed in previous work. Numerical simulations further validate the theoretical analysis, regardless of the random initial state, demonstrating that the system consistently converges to a unique positive equilibrium. From a therapeutic perspective, we propose targeted therapeutic strategies and verify their effectiveness through numerical simulations. These results provide a universal theoretical basis for understanding dynamic mechanisms of Alzheimer's disease and offer critical guidance for developing targeted therapeutics.

We provide a systematic and rigorous geometric framework that relates three structures naturally associated to BPS central charges in $\mathcal{N}=2$ supersymmetric gauge theories: the split attractor flow (SAF) of $|Z|$, the Hessian flow (HF) of $\operatorname{Im}(e^{-i\vartheta}Z)$, and the spectral network (SN) on the base curve of the Hitchin fibration. Our main contributions are: (i) a concise proof of orthogonality between SAF and gradient Hessian flow using only the Kähler structure; (ii) a precise lift–projection duality showing that the spectral network projects to the characteristic Hessian flow (the Hamiltonian flow of $\operatorname{Im}(e^{-i\vartheta}Z)$) on the Hitchin base, clarifying a crucial distinction; (iii) a complete proof of the Kontsevich–Soibelman (KS) equivariance by induction on the SAF tree depth, with the geometric ordering provided by the characteristic Hessian flow. We illustrate the framework with detailed and nontrivial examples: $SU(2)$ pure and $N_f=4$ (including BPS indices for higher flavour charges), $SU(3)$ pure (full BPS spectrum reconstruction), $SU(4)$, the Kronecker $3$-quiver, and we apply the induction to derive a closed-form BPS spectrum for the Argyres–Douglas $H_1$ theory, $\Omega(n\alpha_1+m\alpha_2)=\frac{1}{n+m}\binom{n+m}{n}\binom{n+m}{n+1}$, which is known from Cecotti–Vafa and serves as a strong consistency check of our geometric recursion. In the tropical limit we obtain an explicit generating function for disk counts in $SU(N)$ gauge theories, $Z_{\mathrm{disk}}^{SU(N)}(y) = \exp\!\,\Bigl( \sum_{\alpha\in\Phi_+} \sum_{k=1}^{\infty} \frac{1}{k}\binom{k+\mathrm{ht}(\alpha)-1}{\mathrm{ht}(\alpha)-1} e^{-k\langle\alpha,y\rangle} \Bigr) $, which reproduces the standard scattering diagram result and confirms the geometric framework.

A recent result due to Teng Zhang compares the sum of $m$ matrices and the sum of their quadratic symmetric moduli: $$ \left\| \sum_{k=1}^m A_k\right\| \le \sqrt{2} \left\| \sum_{k=1}^m |A_k|_{\qsym}\right\| $$ for every unitarily invariant norm. Here $|A|_{\qsym}$ is the quadratic mean of $|A|$ and $|A^*|$. We derive operator and eigenvalue refinements of Zhang's inequality from a new polar decomposition for the quadratic symmetric modulus. For instance, $$ \left| \sum_{k=1}^m A_k\right| \le \frac{\sqrt{2}}{2} \left\{ \sum_{k=1}^m \left(|A_k|_{\qsym}+V|A_k|_{\qsym}V^*\right)\right\} $$ for some unitary matrix $V$. We also establish the polar decomposition for the maximal modulus associated with Olson's order, and derive, as in the quadratic case, a series of estimates.

We provide algorithms that compute $\epsilon$-estimates of the $\ell_p$-Lewis weights of a matrix $A \in \mathbb{R}^{m \times n}$ for $p \geq 4$ using $O(p^2 \log(m/\epsilon))$ rounds of leverage score computation, where $\ell_p$-Lewis weights and leverage scores are both standard measures of row importance. This improves upon the state-of-the-art round complexity of $O(p^3 \log(m/\epsilon))$ due to Fazel, Lee, Padmanabha, and Sidford (2022). We obtain our results by carefully applying a local variant of relatively smooth gradient descent to primal and dual forms of the $\ell_p$-Lewis weight optimization problem and providing tools to convert between different notions of approximate $\ell_p$-Lewis weights.

Let $S_N=\{1\le d\le N:d=x^2+y^2\text{ for some }x,y\in\mathbb Z\}.$ We prove a power-saving form of the van der Corput property for $S_N$. As a consequence, we obtain a strong Sárközy-type result: if $A\subseteq [N]$ has no nonzero difference equal to a sum of two squares, then $|A|\ll_\varepsilon N^{7/8+\varepsilon}$ for every $\varepsilon>0$, improving upon an earlier quasipolynomial bound due to Rice. The shape of this bound is optimal, as a construction of Younis yields a set $A\subseteq [N]$ with $|A|\gg N^{1/2}$ such that $(A-A)\cap S_N=\emptyset$.

The existence of a Moore graph of degree $57$ is the last open case in the Hoffman-Singleton classification of Moore graphs of diameter two. We prove that such a graph, if one exists, has no involutory automorphisms; consequently its automorphism group has odd order. The proof relies on a new trace-rank identity, valid for every finite graph and obtained from the Brauer quotient of $p$-permutation lattices: the trace of an automorphism of prime order $p$ on a spectral summand whose projection idempotent preserves $p$-adic permutation lattice equals the rank, over the residue field of characteristic $p$, of the corresponding idempotent restricted to the fixed vertices. For strongly regular graphs, this upgrades the classical character-value congruences for prime-order automorphisms to exact rank equalities. As further applications, the identity yields new restrictions on automorphisms of odd prime order of the hypothetical Moore graph, and it sharpens known automorphism analyses of other hypothetical strongly regular graphs.

A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation. Adam's per-coordinate preconditioner drifts along each symmetry orbit, which pulls the trajectory off the symmetry quotient where the optimization lives and blurs the singular-learning rate the quotient makes readable. We build DDC, a Dead-Direction Conditioner that lifts a base optimizer into a $G$-equivariant one: it conditions the optimizer's state in the orbit decomposition of a $G$-invariant metric, so the trajectory stays a preconditioned gradient flow on the quotient $\bar\Theta = \Theta/G$. The construction carries four architectural gauges (cross-entropy shift, ReLU and SwiGLU rescaling, LayerNorm and RMSNorm scale, and a per-head $O(d_{\rm head})$ attention rotation matched to RoPE), proves exactly equivariant on an Adam base, and composes with a Muon base through a gauge-equivariant orthogonaliser. Respecting the symmetry changes both the minimum the optimizer reaches and what it leaves measurable there. On a language model trained past the point of fit, DDCAdam resists the over-training collapse AdamW falls into, holding a validation-train loss gap of 0.67 against 5.88, and reads the dead-direction rate in 32 of 65 layer-by-observable cells where AdamW reads it in 7. A vision transformer trained from scratch reaches lower validation loss (1.71 against 2.12) while compressing spare feed-forward capacity a matched AdamW leaves intact. On a Muon base, where the rotation gauge composes exactly, DDCMuon groks ten of eleven seeds at depth 24 that a plain Muon never reaches. Built into the optimizer, a network's gauge symmetry sharpens the minimum it finds and turns that minimum's geometry into something the trajectory can measure.

We develop a unified framework for studying the integers missing between consecutive terms of an increasing integer sequence, extending Barry's arithmetic gap-sum to geometric and harmonic analogues via the theory of quasi-arithmetic means. All three gap-sums admit a common interpretation: each equals the gap size multiplied by the appropriate mean of the missing integers. Building on this, we prove a general sparse summation theorem expressing the sum of a strictly monotonic function over a sparse integer sequence as the full range sum corrected by the gap-sums of the missing portions. Specializing on the three Pythagorean means recovers a classical formula of al-Kāshī from the fifteenth century in the arithmetic case, and yields explicit formulas in the geometric and harmonic cases. As a concrete application of the geometric case, we derive a product identity involving the Fuss–Catalan numbers. Applying the harmonic case to the Fibonacci and Lucas sequences, we establish that the harmonic gap-sum converges to $\ln(\alpha)$ exponentially, where $\alpha$ is the golden ratio, and derive explicit two-term asymptotic expansions for the tails of the reciprocal Fibonacci and Lucas series with closed-form coefficients, and establish the asymptotic formula $H_{u_n} \sim n\ln(\alpha)$ for both $u_n = F_n$ and $u_n = L_n$, with explicit $O(1)$ error terms that differ due to their distinct initial conditions. As a further consequence, by comparing the gap-sum expansions with the classical Hardy–Wright expansion of harmonic numbers, we derive exact series identities expressing Euler's constant $\gamma$ in terms of harmonic numbers at Fibonacci and Lucas indices, and obtain a new identity relating the reciprocal Fibonacci constant $\psi$ and the reciprocal Lucas constant $\psi_L$.

We study the two-dimensional incompressible viscous non-resistive magnetohydrodynamics in the periodic strip $\mathbb T\times\mathbb R$, subject to a smooth sheared background magnetic field $(\xi(x_2),0)^{\top}$, where $\xi(x_2)$ is bounded and away from zero. For sufficiently smooth perturbations satisfying even-odd symmetry, we prove global-in-time well-posedness and nonlinear stability in Lagrangian coordinates. The spatial inhomogeneity of the shear profile generates persistent linear contributions, most critically a nontrivial pressure term that precludes the uniform-in-time estimates. We straighten the integral curves of the initial magnetic field and construct a volume-preserving corrector. This geometric reduction transforms the intractable linear pressure into a quadratic nonlinearity. These structures yield the global energy bounds and the anisotropic algebraic decay rate for the system. This mechanism appears to provide the first rigorous framework for establishing global nonlinear stability for viscous non-resistive magnetohydrodynamics near the genuinely nonuniform sheared magnetic profile.

We introduce a new elementary technique for proving $\#P$-hardness results under Turing reductions, and showcase its applicability by settling the counting complexity of several fundamental problems about counting orderings in graphs, posets, and geometric configurations. The technique addresses a natural difficulty in counting reductions: when designing gadgets that attempt to enforce a semantic property, the introduction of the gadgets themselves can affect the function to count in complicated ways. In a nutshell, we introduce an amplification parameter $q$ for the gadgets (e.g., replacing a subset of the vertices of a graph by $q$ copies) and show that the limiting behavior when $q$ tends to infinity, or even to negative values, filters out the unwanted effects of the gadgets, which allows us to recover the desired counts. These limits can be computed symbolically via interpolation, after showing that the probability of a random ordering satisfying the problem's constraints is a rational function in $q$. Concretely, with this methodology, we show $\#P$-completeness for counting the following objects: (a) successive vertex orderings, even on bipartite graphs, (b) $st$-numberings of graphs, (c) shellings, even for bipartite graphs, (d) perfect elimination orderings, even for split graphs, (e) linear extensions of N-free posets of height $3$, (f) linear extensions of posets of height $2$, and (g) peeling sequences of point sets. Result (e) settles a conjecture of Felsner and Manneville (2015), and although result (f) was first proved by Dittmer and Pak (2018), we include an alternative proof, using our technique, that does not rely on the result of Brightwell and Winkler (1991) about hardness of counting for general posets. Result (g) answers a question of Eppstein.

We introduce an operator-theoretic framework for long-range operators over general dynamical systems with analytic hopping and small potential. By establishing a partially hyperbolic splitting on the fibered solution bundle, we define the Canonical Center Bundle (CCB) as the center subbundle of this splitting, which is shown to be globally trivial. The center bundle admits a representation via Riesz spectral projections of the transfer operator. Furthermore, we show that, in the local regime, the center bundle arising in this framework essentially coincides, in the sense of gap convergence, with the Intrinsic Center Bundles (ICB) obtained from finite-range approximations in \cite{GJ}. The partially hyperbolic structure thereby reduces the spectral problem to the center bundle, leading to a Johnson-type characterization of the spectrum in terms of the associated center cocycle. We then apply this framework to quasi-periodic Schrödinger operators with analytic hopping, large analytic potentials and Diophantine frequency. In this setting, the center cocycle is analytic and satisfies a Center Thouless formula. As consequences, we establish the absolute continuity of the integrated density of states (IDS), resolving a problem of Eliasson; prove quantitative Hölder continuity of the IDS, partially answering a question of You; and obtain Anderson localization for the original Schrödinger operators.

We present an a priori error analysis of consistent-loss PINNs for stationary convection-diffusion equations on curved level-set domains. The standard mean-squared interior loss fails in the convection-dominated regime: the solution develops an $O(\eps)$ boundary layer in which the pointwise residual grows like $\eps^{-1}$, so the loss is dominated by the few collocation points inside the layer and leaves the smooth bulk unresolved. We remove this mismatch by penalising the interior residual in a discrete $\Lp{\gamma}$ norm with $\gamma = 1 + 1/\log\mtil$, a computable surrogate for the $\Hminusone$ stability term, and imposing the boundary condition through a discrete $\HhalfBdry$ trace norm, which treats flat and curved geometries uniformly. Under Besov regularity assumptions we prove a single a priori $\Hone$ error bound, valid for all interior exponents $\gamma \in (1,2]$, with an optimal recovery rate governed by a cut-cell floor $1/(2\gamma)$ specific to the curved geometry. Numerical experiments on a rectangle and a disk at $\eps = 2^{-s}$, $s \in \{2,4,6\}$, confirm the analysis: as the layer sharpens, the $\Lp{2}$ interior loss becomes seed-fragile while the $\Lp{\gamma}$ interior trains reliably, the interior norm being the decisive factor in convergence.

In this paper, based on the concept of weighted distribution, we introduce a kind of new approximations for sums of lognormal random variables, such that they are both comonotonic and moment matching. Numerical results show that the approximation performance of the newly presented approximations is, overall, comparable to the classical comonotonic approximations, but in terms of the right tail of the distribution of the original sum our approximations perform better than the classical comonotonic ones. Another contribution of this article is the establishment of the step-weighting theory for continuous random variables.

In this paper, we prove Gromov's simplicial volume vanishing conjecture for closed manifolds with spin universal cover. More precisely, we show that if a closed oriented manifold admits a metric of nonnegative scalar curvature and its universal cover is spin, then its simplicial volume vanishes. In particular, a closed oriented aspherical manifold with nonzero simplicial volume admits no metric of nonnegative scalar curvature.

This paper presents a generic approach for establishing mean field game master equations, applicable whenever the mean field equilibrium can be characterized by a McKean-Vlasov forward-backward stochastic differential equation system. The core of our approach is a representation formula for the first-order Lions derivative of the decoupling field of this forward-backward SDE system. We then employ a bootstrap argument to recursively compute its higher-order derivatives. To demonstrate the method's versatility, we establish the local well-posedness for master equations in three distinct models: extended mean field games, mean field games with volatility control, and mean field games with a major player.

In this paper, we establish higher-order convergence rates of the periodic homogenizatio for symmetric Lévy-type operators, encompassing the subcritical $\alpha$-stable regime, critical regime, and supercritical diffusive regime. To this end, we develop a systematic framework to decompose the contributions of the underlying jumping kernel across small, intermediate, and large spatial scales – a strategy tailored to all the aforementioned regimes. To the best of our knowledge, this work represents the first comprehensive study of higher-order convergence rates in the homogenization of non-local operators.

Ozaki scheme II emulates high-precision matrix multiplication using low-precision integer matrix operations based on the Chinese remainder theorem (CRT). It first scales the high-precision matrices to convert them into integer matrices. For this scaling step, Ozaki scheme II provides two modes: accurate mode, which uses INT8 matrix multiplication to estimate scaling factors, and fast mode, which applies the Cauchy–Schwarz inequality at lower computational cost. We show that the existing formula lacks scale invariance; multiplying the input matrices by a constant changes the effective bit width of the integer matrices in the scaling step, causing accuracy degradation or CRT recovery failure. To address this, we propose a revised scaling formula derived from the CRT uniqueness condition via the Cauchy–Schwarz inequality. The proposed formula is scale-invariant by construction, guarantees that the CRT uniqueness condition is always satisfied, and introduces no additional overhead over the original fast mode. Experiments on an NVIDIA GH200 GPU show that the proposed method achieves accuracy comparable to that of accurate mode while maintaining throughput comparable to that of fast mode. In the accuracy–throughput trade-off, the proposed method overcomes the accuracy limitation of fast mode and the throughput constraint of accurate mode, offering a superior accuracy and performance.

We prove that epsilon multiplicity can take transcendental values. The main structural result is a one-ideal formula for section rings: under natural positivity hypotheses, the epsilon multiplicity of an ideal generated in one degree is equal to an integral of a divisor-volume function. This formula transports an asymptotic colength invariant of ideals to the geometry and arithmetic of divisor volumes. To produce a transcendental value, we combine the formula with a shifted projective-bundle construction inspired by Bornträger and Nickel. The shift places the construction in the positivity range required by the one-ideal formula while preserving the underlying disk geometry of the volume computation. Reversing the order of integration reduces the resulting integral to three integrals of rational functions. Their arctangent terms cancel exactly, whereas the remaining real logarithms form an explicit algebraic linear combination whose value is positive. Baker's theorem then implies transcendence. Consequently, there exists a homogeneous ideal in a normal standard graded domain whose epsilon multiplicity is transcendental.

We construct, for every $g\geq 2$, infinite families of homotopic smooth embeddings of a closed genus-$g$ surface whose images are pairwise not smoothly image-concordant, while each surface is $\pi_1$-injective. The main closed examples lie in one-fold stabilizations of closed aspherical mapping tori with torsion-free fundamental group: after stabilization by $S^2\times S^2$, the surfaces have a common framed dual sphere and the inclusion of each complement induces a $\pi_1$-isomorphism. The image-nonconcordance already occurs before stabilization, in the underlying closed aspherical mapping torus, and persists after every finite number of $S^2\times S^2$-stabilizations. The obstruction is a computable mod-two coordinate of Freedman–Quinn/Dax-type self-intersection data for concordance tracks, indexed by self-dual double-cosets of a possibly non-normal surface subgroup $H\leq\pi_1X$. The geometric source of the relevant labels is a M"obius-band square-root relation: elements $t\notin H$ with $t^2\in H$ produce self-dual labels in torsion-free ambient groups. These square roots are realized naturally in Klein-bottle $I$-bundle pieces and persist in closed graph-manifold mapping-torus examples.

In this paper, we study pure-projective tilting modules and related classes of rings. We introduce the notion of a pure-tilting hereditary ring, namely, a ring over which every ideal is pure-projective tilting, and investigate its structural properties. We prove that a ring R is a pure-tilting hereditary ring if and only if R is hereditary noetherian over a von Neumann regular ring R. In the commutative case, we show that R is a pure one-tilting hereditary ring precisely when R is hereditary noetherian. Using Kaplansky conjecture, we establish a connection between pure-tilting hereditary rings and the hereditary noetherian property of prime factor rings. In category theory, for the torsion pair consisting of Gen of I and the orthogonal class of I in the category of R-modules, we establish that the associated Happel-Reiten-Smalo heart H sub I is a Grothendieck category. We also examine the characterization of Ext-orthogonal classes determined by pure projective tilting modules. In addition, we show that every Gorenstein pure projective tilting module is Gorenstein flat if and only if every Gorenstein pure projective tilting module is strict T-stationary, where T denotes the class of all finitely presented tilting modules. These results establish new links between tilting theory, hereditary ring conditions, and Gorenstein homological structures.

Graph-based narrative extraction relies on a coherence function to score transitions between events, but the coherence metrics in current use are defined operationally and lack an information-theoretic foundation. We study the composite metric $C=\sqrt{A\cdot T}$, where $A$ is the angular similarity of document embeddings and $T=1-d_{\mathrm{JS}}$ is a topic proximity from the Jensen-Shannon distance of soft memberships, and give it an information-geometric reading together with an axiomatic characterization of the geometric-mean combinator. On the product manifold $\mathbb{S}^{d-1}\times\Delta^{K-1}$, the negative log-coherence decomposes additively into an angular and a topic cost. Because the Riemannian metric tensor induced by the Jensen-Shannon distance on the simplex is proportional to the Fisher information matrix, the topic component is locally consistent with the Fisher-Rao metric singled out by Chentsov's theorem. Within the compensability spectrum of combinators, the geometric mean is the unique one consistent with four natural axioms (a boundary/veto condition, symmetry, log-additivity, normalization), and the construction motivates a proper product metric $d_\times$. Experiments on four corpora, three embedding families, and three topic models are consistent with the framework: the Fisher identity holds ($R\ge0.99$), the geometric mean tracks $d_\times$ closely ($\rho=0.999$), and a downstream LLM-as-judge check finds it is not dominated by any alternative combinator or single-channel baseline. Sweeping the spectrum, the bottleneck-coherence gap between extracted and random storylines splits into a symmetric component, maximized at the geometric mean across five corpora, and a displacement term; a cross-modal image-narrative case study reproduces the effect. These results justify the composite coherence metric and articulate when the geometric mean is the natural choice.

Extensible lattices are a common choice of nodes for approximating multidimensional integrals by a sample mean, typically using sample sizes of the form $n = b^m$ for a prime base $b$ and non-negative integer $m$. However, a computational time budget or early termination due to hardware failure may prevent us from choosing the preferred number of samples. In this paper, we establish an upper bound on the figure of merit $P_\alpha$ for extensible lattice sequences \emph{with arbitrary $n$}, derived as a worst-case error in a Banach space setting. We show that although the upper bound on $P_\alpha$ can decay no faster than $\mathcal O(n^{-1})$ for general $n$, restricting $n$ to the form $n =\lambda b^p$, where $\lambda$ is a fixed integer, we can recover the optimal decay of nearly $\mathcal O(n^{-\alpha})$ as $p \to \infty$. We also investigate a related figure of merit, $P_{\alpha,2}$, arising as a worst-case error for Hilbert spaces of integrands. We obtain a theoretical upper bound on $P_{\alpha,2}$ in terms of $P_{\alpha}$ for equal sample weights and show numerically that optimal sample weights can improve the cubature error bound even further.