Papers

We present the discrete–complex complement quotient (DCCQ) framework through the ordered Bernoulli-word kernel $f(p,n,k)=p^k(1-p)^{n-k}$ and its inverse-integer level sets. The binary level $2^{-n}$ selects $p=1/2$ as the unique real split-independent anchor. Imposing the additional continuation rule $1-z=\overline z$ gives the conjugation-symmetric line $z=1/2+iu$. On the central normalized branch, the inverse-level split $\kappa=k/n$ also has real part $1/2$. The quadratic coordinate $Q(z)=1/4+u^2$ identifies complementary conjugate points, has minimum $1/4$, and has an explicit two-sign inverse; restricting its values to integers defines an arithmetic quantization. For supplied positive critical-line zero ordinates $\gamma_k$, the induced level has the exact decomposition \[ L_k=\frac14+\gamma_k^2=N_k+\delta_k,\qquad \delta_k=\widetilde B_1\!\left(\gamma_k^2+\frac34\right), \] where $N_k=\lfloor L_k+1/2\rfloor$ and $\widetilde B_1(x)=\{x\}-1/2$. The integer–phase pair $(N_k,e^{2\pi i\delta_k})$ retains the level and the positive ordinate. Prime-exponent coordinates describe the factorization of the integer component, while conditional Fourier moments formulate an open question about the residual phases. The finite numerical diagnostics are descriptive. A distinct complex exponent $s$ gives an inverse-level representation of the supplied term $m^{-s}$; direct substitution cancels the auxiliary coordinates. These terms assemble into the classical Dirichlet series and Euler product for $\Re s>1$. The framework organizes exact coordinate identities, classical zeta connections, and numerical diagnostics; it establishes neither conditional equidistribution nor a zero-location theorem.

In 2023, the first author, Nunez, Singh, and Swain [10] formulated an analogue of Kaplansky's idempotent conjecture for integral quandle rings. In this paper, we develop a new Fourier-analytic approach to quandle rings and use it to establish the conjecture for two major classes of quandles: Takasaki quandles, including dihedral quandles, and medial commutative quandles. A central contribution of the paper is the introduction of Fourier analysis on Alexander quandles, which provides a new framework for studying quandle rings and, in particular, their idempotents. Using this Fourier-analytic framework, we also prove that a previously known sufficient condition for the existence of counterexamples to the conjecture is in fact necessary, thereby resolving a problem of Jablonowski [12].

We consider the space of $L^p$-bounded martingales on the Cantor space for $p\in[1,\infty]$. Equipping the Cantor space with two standard metrics, we show that under both, a typical martingale diverges at a typical point. By a 'typical martingale', we mean an element of a co-$\sigma$-porous set when $p\in[1,\infty)$ and of a co-porous set when $p=\infty$. Specifically, we show that while a typical martingale diverges on a co-$\sigma$-porous set with respect to the first metric, no martingale diverges on a co-$\sigma$-porous set with respect to the second metric. Finally, we investigate the key factors determining whether the divergence set can be co-$\sigma$-porous.

We introduce a new approach to the study of Bochner-Kähler manifolds, based on normal form techniques in CR geometry. As observed by Webster, Bochner-Kähler manifolds are intimately related to spherical Sasakian manifolds. Our approach utilises this relationship as well as normal forms for spherical Sasakian manifolds (known as rigid spheres). In this setting potentials of Bochner-Kähler manifolds are nothing but the defining equations of rigid spheres in rigid normal form. This leads to a canonical formula for Bochner-Kähler potentials, as well as streamlined proofs of known results on the structure and the symmetries of Bochner-Kähler manifolds. Our approach also provides a host of examples in terms of an implicit formula of the corresponding Kähler potentials.

In the present paper, we consider the maximal operator \[ \mathcal M_{\Omega}f(x) := \sup_{r>0}\frac{1}{r^n} \int_{|y|<r} |f(x-y)| \left| \Omega\!\left(\frac{y}{|y|}\right) \right|\,dy. \] A longstanding open conjecture raised by E.M.Stein asks whether the maximal operator $\mathcal M_{\Omega}$ is of weak type $(1,1)$ when $\Omega$ is merely in $L^1(\mathbb S^{n-1})$. We partially settle this problem by proving weak type $(1,1)$ bounds of $\mathcal M_{\Omega}$ with kernel $\Omega\in\mathcal X(\mathbb S^{n-1})$, yielding a significant improvement over the work of M.~Christ and Rubio de Francia. Here$\mathcal X(\mathbb S^{n-1})$ is the space related to the Hausdorff–Choquet angular space and $$ L\log^+\!L(\mathbb S^{n-1}) \subsetneq \mathcal X(\mathbb S^{n-1})\subset L^1(\mathbb S^{n-1}). $$ Prior to our work, the condition $\Omega \in L\log^+\!L(\mathbb S^{n-1}) $ remained the best known size restriction guaranteeing the weak-type$ (1,1)$ bounds for all dimensions $n > 2$. To our knowledge, the present paper is the first to relax this threshold for $n>2$. Finally, for the general Hausdorff–Choquet scale $\mathcal H\mathcal C_{\alpha}$, we show that $\alpha=(n-1)/2$ is the sharp exponent for uniform weak type $(1,1)$ estimates.

Let $m\geq 2$ and $d\geq 7$. We consider homogeneous polynomials in $2m+2$ variables of the form $f=F_0(u_0,v_0)+\cdots+F_m(u_m,v_m)$, where the $F_i$ are independently very general squarefree binary forms of degree $d$. We prove the rational noncommutative Hodge conjecture for the dg category $\mathrm{MF}^{\mathrm{gr}}(f)$ of graded matrix factorizations. Its Hochschild homology is the direct sum of the scalar-invariant Jacobian sector and $d-1$ one-dimensional point sectors. Boundary–bulk images of explicit rank-one factorizations generate a lattice of rank $(d-1)^{m+1}$ in the identity sector, while grading shifts of the stabilized residue field generate all point sectors. A reduced-Burau calculation shows that the identity-sector lattice exhausts the rational Hodge classes at a very general parameter. Consequently, $\dim_{\mathbb{Q}}\operatorname{Hdg}\bigl(\mathrm{MF}^{\mathrm{gr}}(f),\mathbb{Q}\bigr)=(d-1)^{m+1}+d-1$, and the rational topological $K$-rank is $\bigl((d-1)^{2m+2}+d-1\bigr)/d+d-1$. Finally, applying the additivity of the noncommutative Hodge conjecture for semiorthogonal decompositions together with Orlov's decompositions for Fano, Calabi–Yau, and general-type hypersurfaces proves the rational Hodge conjecture for the associated smooth projective hypersurface.

We analyse magnetic Robin Laplacian in curvilinear half-plane with a smooth boundary. It is well known that the spectrum of the Robin Laplacian is unstable with respect to boundary deformations. This means that if the boundary is a straight line then the spectrum of the Robin Laplacian is purely essential. From the other hand, the perturbation of the boundary produces eigenvalues below the essential spectrum. In this paper, the Robin-Laplace operator with a compactly supported magnetic field is considered. We prove that the spectrum of the magnetic Robin Laplacian is stable under small and local deformations of the boundary.

This paper addresses the numerical solution of nearly incompressible linear elasticity boundary value problems and their associated eigenvalue problems. A mixed finite element formulation based on Mini element is proposed to circumvent the locking phenomenon that plagues standard low-order elements in the nearly incompressible limit. For the boundary value problem, we establish the well-posedness of mixed variational formulation and derive a priori error estimates that are uniform with respect to the Lamé constant $\underline{\lambda}$, thereby proving the method's locking-free property. For the eigenvalue problem, we develop an efficient non-nested augmented subspace algorithm designed within the mixed finite element framework. A comprehensive convergence analysis is provided for the discrete eigenvalue approximation and the proposed iterative solver, demonstrating that the convergence rates remain independent of $\underline{\lambda}$. Numerical experiments on both problems confirm the theoretical conclusions, showing optimal convergence rates and robustness as $\underline{\lambda} \to \infty$. The results validate the effectiveness of Mini element and proposed augmented subspace algorithm for reliable and efficient computation in the nearly incompressible regime.

We prove that the algebra of multiple Eisenstein series is an $\mathfrak{sl}_2$-algebra. In particular, we show that it is graded by weight, which also holds after taking complex linear spans. Further, we identify it as a graded $\mathbb{Q}$-algebra with the associated graded algebra of $q$-analogues of multiple zeta values with respect to the weight filtration. The main ingredient is an estimate that compares the usual defining sums of multiple Eisenstein series with sums over a lattice order depending on an integer $N$.

As sixth-generation (6G) communication systems evolve toward higher frequency bands and larger array apertures, the near-field range expands rapidly, making near-field channel estimation increasingly important and challenging. Beam training has been recognized as an effective approach for channel state information (CSI) acquisition. However, because of the propagation characteristics of spherical waves, beam training needs to perform a joint search in the angle and distance domains, which results in unaffordable beam training overhead. By flexibly reconfiguring antenna positions, movable antenna (MA) technology can fully exploit the spatial variations of wireless channels and achieve more accurate beam focusing, thereby providing additional flexibility for efficient beam training design. Therefore, based on MA-assisted near-field systems, we develop a hierarchical beam training strategy that combines reduced training overhead with high beam gain and design a corresponding hierarchical codebook. This codebook forms focused beams over the joint angle-distance domain, maximizing beam gain within the target region while suppressing energy leakage into non-target regions. Simulation results confirm substantial performance gains of the method over the conventional fixed-position antenna (FPA) system and the far-field beam training method.

We prove that the square root of the separable quantum optimal transport cost associated with the orthogonal projection onto the antisymmetric subspace defines a genuine distance between density matrices. Equivalently, this establishes the triangle inequality for the order-two Beatty-França quantum optimal transport construction induced by the Hilbert-Schmidt distance between pure states. The result also proves metricity of the corresponding distance derived from separable SWAP fidelity. The proof replaces the unavailable gluing argument by convex-roof duality and a dimension-independent interpolation result for Hermitian operators.

Grid-scale battery energy storage can generate revenue by stacking services across electricity and frequency response markets, yet identifying the lifetime profit-maximising stacking strategy remains challenging. Decisions across services are coupled through shared battery system capacity, constrained by system operator energy management rules, and further shaped by product-specific technical requirements that govern system operation, degradation, and lifetime profitability. This paper presents an ageing-aware receding-horizon framework for co-optimising multi-service stacking that explicitly captures product-specific characteristics and state-of-energy compliance rules. The framework is applied to the Great Britain market, where storage operators can stack electricity trading with multiple dynamic frequency response services procured through the newly introduced 'Enduring Auction Capability' platform under energy management requirements imposed by the National Energy System Operator. Using real market data, we demonstrate that degradation modelling, discount rate, and battery ageing jointly govern both lifetime value and optimal stacking strategy. Accounting for ageing increases lifetime revenue by up to 32% relative to a degradation-agnostic benchmark, while higher-fidelity ageing modelling can provide a further revenue improvement of up to 16% over simpler formulations. Lower discount rates favour strategies that balance calendar and cycling ageing, while higher discount rates favour aggressive operation. Across all strategies, services responding to positive frequency deviations are consistently preferred over those responding to negative deviations. Lifetime profit is maximised by adapting the optimal service mix as the battery ages.

The computation of the Bures-Wasserstein (BW) barycenter of an ensemble of positive definite matrices arises throughout machine learning, optimal transport, and quantum information. Riemannian gradient descent (RGD) at unit step size – the fixed-point iteration used in practice – converges rapidly, yet existing analyses present a dichotomy: unit-step guarantees carry worst-case exponential dependence on the dimension, while dimension-independent guarantees require small step sizes that forfeit the empirical speed. We resolve this dichotomy, not by improving the guarantees for unit-step RGD, but by proposing a Projected RGD algorithm that achieves dimension-independent linear convergence at unit step size. The achieved rate, $(1 - \kappa^{-3/2})$, where $\kappa$ is the condition number of the ensemble, also polynomially improves on the best small-step guarantee ($\kappa^{3/2}$ versus $\kappa^{5/2}$ iteration complexity). The crux is a novel Projection Lemma: clipping the eigenvalues of a positive matrix to an interval $[\alpha, \beta]$ is the closed-form, non-expansive (1-Lipschitz) BW-metric projection onto the set $\{S : \alpha I \leq S \leq \beta I\}$ – a statement which, unlike its known one-sided counterpart, does not follow from convexity. The projection is moreover free: it reuses an eigendecomposition the next iteration must perform in any case, so the projected and unprojected iterations cost the same per step. The same analysis covers the invariant matrix projection problem of Brahmachari et al. (2025), whose fixed-point algorithm we identify as unit-step RGD on a totally geodesic submanifold, thereby extending the dimension-independent guarantee to that setting verbatim.

For planar graphs, it is known that their treewidth is bounded by $O(\sqrt{n})$, where $n$ is the number of vertices of the graph. A related invariant to treewidth, is the scramble number of graphs. Recently, Connor et. al proved that planar graphs of bounded maximal degree have scramble number bounded by $O(\sqrt{n})$. An open question is whether the scramble number of any planar graph follows this same bound. We give a definitive answer with an explicit bound for a subset of planar graphs, the simple outerplanar graphs and the simple near outerplanar graphs.

The objective of the resource-constrained project scheduling problem (RCPSP) is to minimize makespan while satisfying precedence and renewable-resource constraints. Priority-rule heuristics construct feasible schedules with low computational cost and explicit decision logic, making them widely used in practice and an attractive alternative to more computationally intensive methods. However, no traditional rule performs consistently well across projects, and researchers have therefore investigated automated priority-rule design. Genetic programming (GP) hyper-heuristics have been the predominant approach to this task, but evolving a high-performing rule may require evaluating many candidate rules on the training projects. Recent advances in large language models (LLMs) make it possible to generate and iteratively revise priority rules using performance feedback. This paper presents an LLM-guided population-based framework for automated priority-rule design. During the offline search, an LLM generates and revises candidate rules, while schedule quality on training projects determines candidate fitness and guides subsequent revisions. At the end of the search, the best rule is returned and applied directly to unseen projects without further search or LLM calls. Experiments show that the LLM-designed rules outperform traditional single rules across the test sets and outperform GP-designed rules obtained under comparable search effort, while remaining competitive with rules obtained from a substantially larger GP search. On large projects, selected LLM-designed rules outperform all considered traditional rules and two genetic algorithm configurations on highly parallel projects. An ablation study examines how changes to the main search components affect performance, while rule analyses describe the structure and decision behavior of the LLM-designed rules.

Let $\Omega\in L^1(S^{d-1})$ have vanishing average, and let $T_\Omega^\ast$ be the maximal truncation of the associated rough homogeneous singular integral. We prove quantitative sparse bounds for $T_\Omega^\ast$. If $\Omega\in L^\infty(S^{d-1})$, then, for every $1<p<\infty$, \[ \|T_\Omega^\ast\|_{(1,p)\text{-}\mathrm{sparse}} \lesssim_d p'\|\Omega\|_{L^\infty(S^{d-1})}. \] For unbounded angular kernels, if $1<q<\infty$ and $\Omega\in L^{q,1}\log L(S^{d-1})$, then the same estimate holds for $q'\leq p<\infty$, with the right-hand side replaced by \[ C_{d,q}p' \|\Omega\|_{L^{q,1}\log L(S^{d-1})}. \] These estimates retain a genuine $L^1$ average in the first entry of the sparse form. In the bounded-kernel case, the upper bound has the same linear growth in $p'$ as the known sparse bound for the nonmaximal operator. The unbounded-kernel estimate includes the critical exponent $p=q'$. As a consequence, we obtain weighted weak-type $(1,1)$ estimates for all $A_1$ weights in the bounded-kernel case and for weights in $A_1\cap RH_{q'}$ in the unbounded-kernel case. The proof combines physical-space linearization and microlocal decomposition with localized sparse testing. An amplitude decomposition of the second input, together with the Rademacher–Menshov inequality, yields the quantitative dependence on $p$.

We study the long-term distributional dynamics and small-noise asymptotics of two-dimensional nonautonomous stochastic primitive equations driven by Gaussian and multiplicative Lévy noise, together with moderate deviations for the purely jump model. For sufficiently small noise, uniform moment bounds and an exponentially weighted terminal estimate yield a pullback absorbing family and tightness, while a lower semicontinuous vertical-moment functional preserves admissibility under weak limits. We prove the existence and uniqueness of a pullback measure attractor in the weak topology of probability measures and establish its upper semicontinuity as both noise components vanish. For the jump-driven equation, we establish a moderate deviation principle in $\mathcal D([0,T];H)\cap L^2(0,T;V)$ with speed $a^2(\epsilon)/\epsilon$. The proof combines continuity of the skeleton map with controlled stochastic convergence based on entropy bounds, truncation, martingale estimates, and direct vertical estimates, avoiding Lipschitz continuity of the vertical derivative of the jump coefficient.

We prove a positivity theorem for the universal Plücker coordinates of Karp and Purbhoo when exactly one parameter is negative, with a sharp uniform threshold governed by the largest Plancherel up-transition probability. At the critical specialisation, positivity of the normalised $(2,2)$-coordinate is equivalent to the Octopus inequality, the main technical tool in the proof of Aldous's spectral gap conjecture. By connecting the inequality with Schubert calculus, this answers questions raised by Caputo and Aldous. For a Wronskian with distinct real zeros, we determine the maximal common half-line on which all branching-balanced coordinates are positive semidefinite. We also give Plücker-theoretic proofs of two hypergraph inequalities of Alon, Kozma, and Puder.

In adaptive bitrate (ABR) streaming, the delivery technology behind most Internet video, each title is encoded at several bitrates, forming an encoding ladder of renditions. Each client plays the highest rendition that its network bandwidth can sustain. We show that choosing the ladder is a problem of scalar quantization of the bandwidth distribution. However, this quantization problem is of an unusual kind: the client's logic pins each quantization cell's reproduction value to the cell's left edge, and the distortion measure is a one-sided quality loss rather than a squared error. The constraint reshapes the classical theory. The quality gap of a ladder is the distance between its average delivered quality and the quality limit that infinitely many renditions would attain. For optimal $n$-rendition ladders the gap $\mathcal{G}_n^*$ decays as $\Theta(1/n)$, not the textbook $\Theta(1/n^2)$. Moreover, with large $n$, $n\,\mathcal{G}_n^*\to C_w=\frac{1}{2}(\int\!\sqrt{p\,Q'}\,\,d B)^2$, with the limit attained when the ladder's rates follow the density $\sqrt{p\,Q'}$, which replaces Panter–Dite's $p^{1/3}$; here $p$ is the bandwidth density and $Q$ the quality–rate curve of the content. The law yields closed-form design rules: quantile rung placement, the rendition count $n(\varepsilon)\approx C_w/\varepsilon$ needed to reach tolerance $\varepsilon$, and the economic ladder size $n^*=\sqrt{\lambda C_w/\kappa}$ when a rendition costs $\kappa$ to operate and the quality gap is priced at $\lambda$. The same analysis extends to the design of ladders employing different video resolutions, codecs, perceptual quality metrics, and the two-dimensional adaptation logic of modern web players.

Let $R$ be a commutative semiring with unity $(1\neq0)$, and let $S$ be a proper multiplicatively closed subset of $R$. In this paper, we introduce the notion of quasi $S$-$n$-ideals in commutative semirings. We the study basic property of quasi $S$-$n$-ideals and establish their relationships with quasi $n$-ideals, $S$-$n$-ideals, $S$-prime ideals, and $S$-primary ideals. We also investigate their behavior under localization. Finally, we determine quasi $S$-$n$-ideals in the quotient polynomial semiring $R[x]/\langle x^m\rangle$ and study their natural extensions via the constructions of idealization and amalgamation.

Robust portfolio rules that reconstruct confidence sets after learning need not preserve the evaluator obtained by prior-by-prior Bayesian transport. In the Gaussian model, this discrepancy is summarized by natural-coordinate displacement: inherited transport preserves it whereas fresh reconstruction can replace it. We price evaluator replacement and trace the resulting optimized curvature through endogenous research. Optimized robust value represents protocol regret as a functional Bregman divergence, while a within-vintage rectangular Gaussian benchmark with constant absolute risk aversion (CARA) yields a stopped recalibration tax. In a versioned model-release economy, validated history propagates through a strictly causal network and a same-cycle share of current optimized marginal value feeds back into research supply. The capacity-constrained equilibrium reduces to a scalar equation with protocol-indexed gain \(\mathfrak g_I^P=\lambda\beta_{R,I}(W_R^P)''\). Purely causal validation cannot create a same-cycle unit mode; provenance changes criticality through optimized curvature. For a scalar primitive supplier-score shock \(z\) in direction \(h_I\) and financial outcome \(\mathcal O\), sensitivity factors as \(\omega_{\mathcal O,h,I}/(1-\mathfrak g_I^P)\). Conditional on a smooth equilibrium state and active cell, completion-time information sharply bounds this multiplier when all compatible timing laws are subcritical; no finite uniform bound exists when the timing set reaches the pole.

Colding established a sharp gradient estimate for the Green function on manifolds with non-negative Ricci curvature, which was extended to positive Ricci curvature by Manea recently. We prove almost rigidity of such gradient estimates for positive Ricci curvature. We show that the average of the gradient deficit, namely $\fint (1-|\nabla b|^2)\,dV$, controls the volume deficit in an optimal quantitative manner; here $b$ is the distance-like function defined using the Green function. This implies a quantitative almost rigidity theorem for manifolds with $\ric\ge(n-1)g$ and a gap theorem for Einstein manifolds. We also prove a rigidity theorem for closed 4-dimensional Einstein manifolds by finding a new monotonicity formula.

Granville and Ono proved that there is an $e$-core partition of $n$ for every $n\in\mathbb{N}$ if and only if $e\geq 4$, which translates to a statement about the existence of defect $0$ blocks for symmetric groups in positive characteristic and defect $0$ unipotent blocks of finite general linear groups in positive, non-defining characteristic. Motivated by analogous applications to the block theory of finite classical groups, imprimitive spetses, and cyclotomic Hecke algebras, we prove similar positivity statements about different variants of $e$-cores for charged multipartitions.

In their work on sphere packing and the modular bootstrap, Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini conjectured the exact high-dimensional exponent of the Cohn-Elkies sphere-packing linear program. OpenAI's Chapter 1 subsequently proved their conjecture by establishing that both Fourier sign-uncertainty radii are $(1/\pi+o(1))\sqrt d$. We prove the binary coding analogue: \[ R_D\!\left(\frac12-\frac1\pi\right)=\frac12. \] We also formulate the two Krawtchouk sign-uncertainty problems and determine both of their asymptotics. If $A^{\mathrm K}_{\pm}(n)$ denotes the smallest radius $r$ for which a nonzero Krawtchouk $(\pm1)$-eigenfunction $f$ exists with $f(0) = 0$ and $f(x) \ge 0$ for all $|x| \ge r$, then \[ \frac{A^{\mathrm K}_{\pm}(n)}n\longrightarrow \frac12-\frac1\pi. \] The lower bound proves a mass-concentration principle similar to OpenAI's Chapter 1 for Hamming space. The upper bound, on the other hand, follows the approach of the spherical-code construction in OpenAI's Chapter 2. Gay, Jeronimo, and Liu formulated a hierarchy for binary codes analogous to the spherical-code construction and improved the best known binary coding rate bounds by evaluating the first level of the corresponding hierarchy. We prove this hierarchy bounds the Delsarte program and give a construction at arbitrarily deep levels of the hierarchy, attaining the upper bound in the limit. The construction uses an $N$-qubit generalization of the pure-state channel of Alrabiah and Guruswami.

For resource-constrained wireless edge devices over bandwidth-limited fading channels, wireless image transmission using traditional separate coding suffers from the cliff-effect collapse. Prevailing deep joint source-channel coding (JSCC) based on convolutional neural networks can mitigate this issue but usually fail to preserve patch-level structures, thereby preventing adaptive per-token power allocation and limiting token-domain compensation for super-resolution (SR). To address these challenges, we propose a token communication framework with SR (TokenComSR). Specifically, we conceive a task-sensitive power allocation (TSPA) module and a signal-to-noise ratio (SNR)-conditioned token refinement module (TRM). TSPA distills training estimates of task sensitivity into inference token power weights, while TRM estimates an SNR-conditioned residual to correct channel-induced distortion in the token domain before decoding. Building on TSPA and TRM, the proposed TokenComSR pairs a Swin Transformer-based token transceiver with a receiver-side SR module for resource-constrained wireless image transmission. Simulation results confirm the effectiveness of the proposed TSPA and TRM, demonstrating improvements over separate coding and JSCC-SR baselines in both reconstruction fidelity and perceptual quality.

In our study, Darbo's fixed point theorem and the coupled fixed point theorem have been extended. The two theorems were generalized using $C-class$ mapping. Using the Darbo type theorem, we gave an existence result for a functional integral equation along with an appropriate illustration. In addition, we explored the numerical solution of a given functional integral problem using modified homotopy perturbation.

This note originated with a mini-course taught at KU Leuven by the author in June 2026. We review the definition of weightings (or quasi-homogeneous structures), the weighted normal bundle, and the weighted deformation space. We emphasize the algebraic approach and functorial properties, using sheaves of ideals and the character spectrum for sheaves of algebras. We review the definition of a multiplicative weighting on a Lie groupoid $G\rightrightarrows M$, and give a self-contained proof of the three-way equivalence between (i) multiplicative weightings of $G$ along $M$, (ii) Lie filtrations of the Lie algebroid $A$ of $G$, (iii) dg weightings of the NQ manifold $A[1]$ along $M$. The corresponding weighted deformation space is a weighted variant of the tangent/adiabatic groupoid and is used in the study of hypoelliptic operators.

We consider a regularly varying stationary sequence of random variables (Xt) with tail index ___ < 2. For these sequences we study the joint convergence of sums, `p- type moduli and maxima. We focus on ratio statistics, including the studentized sums and sums normalized by the corresponding maxima, and study the existence of moments for the limit ratios. We consider particular examples of processes (Xt) whose limit ratios possess all moments. But, in contrast to the latter situation, there also exist sequences (Xt) where certain moments of the limit ratio are in___nite. This phenomenon results from extremal clusters in the sequence.

We study inverse problems for a time-harmonic one-photon model describing the interaction of a single photon with a medium of stationary two-level atoms. After time-harmonic reduction, the unknown compactly supported atomic density appears as a frequency-dependent potential in a scattering equation for the half Laplacian. We prove high-frequency uniqueness results for three types of intensity data: source-driven measurements, renormalized far-field intensity measurements, and phaseless far-field measurements obtained from coherent superpositions of incident plane waves. In each case, the corresponding data, given at all sufficiently large frequencies, determine the atomic density uniquely; the source-driven result requires a geometric visibility condition on the source and observation sets.

We prove that the renormalized volume between the two Epstein surfaces associated with a Weil–Petersson quasicircle is continuous with respect to the Weil–Petersson topology. As a consequence, we establish that the universal Liouville action is proportional to this renormalized volume for every Weil–Petersson quasicircle. We then establish an inequality relating this renormalized volume to the renormalized area introduced by Alexakis–Mazzeo. To this end, we compare both quantities with the Hilbert–Schmidt norm of the associated Grunsky operator, which can be expressed in terms of a nonlocal Lagrangian introduced by O'Hara–Solanes.