We propose novel semi-decoupled and fully-decoupled iterative algorithms for efficiently solving the fully-coupled nonlinear four-field thermo-poroelastic model discretized in space by discontinuous Galerkin method on polytopal grids. We present the model problem, its four-field formulation, and the arbitrary-order weighted symmetric interior penalty scheme exploited for its spatial discretization. Such a scheme is robust with respect to strong heterogeneities in the model coefficients. Then, we present the two solution strategies and prove that under suitable conditions both schemes are convergent. A wide set of numerical simulations is presented to assess the convergence and robustness properties of the proposed method. Moreover, we test the scheme with literature and physically sound test cases for proof-of-concept applications in the geophysical context.

Convex optimization problems with linear equality constraints arise ubiquitously in scientific computing, machine learning, and control theory. Classical Krylov methods are effective but rely on problem-specific preconditioners and high memory. Conversely, gradient-based methods like the augmented Lagrangian method (ALM) avoid these issues yet suffer from slow outer iterations. Developing accelerated outer-iteration schemes, therefore, remains a critical research objective. In this study, we demonstrate that incorporating a proximal operation into the augmented Lagrangian framework yields the proximal ALM, where the outer iteration is equivalent to an implicit gradient descent-ascent (GDA) scheme. We further establish that this equivalence extends naturally to the setting of variable step sizes. Through Lyapunov analysis, we show that the underlying potential function must be shifted from the conventional objective gap to a variational inequality measure, signaling a shift in perspective from pure convex optimization to minimax optimization. Motivated by these observations, we first develop an implicit GDA scheme with variable step sizes based on a continuous-time ODE framework, which achieves an $o(1/k)$ last-iterate convergence rate for both the primal-dual objective gap and the gradient norm. Building upon a second-order ODE framework, we then propose a family of Nesterov-type implicit GDA schemes parameterized by $r \geq 0$, which achieves an $o(1/k^{r+1})$ last-iterate convergence rate for the primal-dual objective gap. Furthermore, specializing the second-order ODE formulation to the case $r=0$, we derive a corresponding explicit GDA scheme and prove an $o(1/k)$ last-iterate convergence rate for the primal-dual objective gap. Finally, we present several numerical experiments to validate these theoretical results and demonstrate the effectiveness of the proposed methods.

In this paper, we develop a continuous-time model-free reinforcement learning algorithm to learn deterministic equilibrium policies in general time-inconsistent control problems. Utilizing the extended Hamilton-Jacobi-Bellman system, we recast the original time-inconsistent problem into an equivalent two-stage problem. In the first stage, for given auxiliary functions, we employ the deterministic policy gradient approach to learn an optimal policy in an auxiliary time-consistent control problem. In the second stage, given the updated policy, we exploit the inner fixed point iterations and some martingale characterizations to learn the auxiliary functions. As a theoretical contribution, we provide some mild model assumptions and establish the convergence of inner fixed point iterations. By repeating this actor-critic style of iterations across two stages, our algorithm aims to learn the equilibrium under different sources of time-inconsistency in a unified manner. The superior effectiveness of the proposed algorithm are illustrated in two classical financial applications with time-inconsistency: mean-variance portfolio management and optimal tracking portfolio under non-exponential discounting.

A new Block Coordinate Gradient (BCG) method, dubbed bAdag, for smooth, nonconvex minimization problem is proposed; it falls in the class of Objective Function Free Optimization (OFFO) methods, and it is based on the AdaGrad algorithm. At each iteration, our method computes an adaptive step size based on the cumulative sum of block gradients, instead of full gradients as in AdaGrad-type methods. We prove ergodic, sublinear convergence rates for the bAdag algorithm when minimizing a smooth, possibly nonconvex objective under the (block) Lipschitz continuity assumption on the gradient. Our theory covers three widely popular block selection strategies: the Cyclic (C) rule, Uniform Random selection (UR), and the greedy Gauss-Southwell (GS) rule. We also extend our algorithm and its convergence theory to box-constrained smooth functions. We validate the proposed algorithms through synthetic and real-world experiments.

Let $L(\lambda)$ be a simple highest weight module of a classical Lie algebra $\mathfrak{g}$ with highest weight $\lambda-\rho$, where $\rho$ is half the sum of positive roots. Joseph proved that the associated variety of the annihilator ideal of $L(\lambda)$ (also called the annihilator variety) is the Zariski closure of a nilpotent orbit in $\mathfrak{g}^*$. Recently, Bai--Ma--Wang introduced a partition algorithm to describe this corresponding nilpotent orbit for a given highest weight module $L(\lambda)$. In this paper, we present a new direct proof of Bai--Ma--Wang's partition algorithm using Sommers duality.

Let $D \subset \mathbb{R}^d$ ($d \geq 2$) be a bounded connected open set of class $C^m$ with $m \geq 3$. We prove that, for a generic domain $\Omega \in \mathcal{O}^m(D)$ in the sense of Baire category for the Micheletti topology, every Dirichlet-Laplacian eigenfunction has nonzero mean, $\int_\Omega \varphi_n \neq 0$ for all $n \geq 1$. This answers a question raised by Steinerberger and Venkatraman. The proof rests on a shape-derivative computation whose degeneracy is identified with an overdetermined elliptic problem of Schiffer type. Rather than resolving this problem, we bypass it through a Baire-category argument based on the meagerness of domains with analytic boundary. As a consequence, we obtain that for a generic domain the Dirichlet heat equation is approximately controllable, and rapidly stabilizable, i.e. stabilizable at any prescribed exponential decay rate, by a single spatially homogeneous scalar control, via the Fattorini-Hautus criterion.

In this work, we settle a problem that dates back to the early 2000s. We provide a complete characterization of special matchings of arbitrary Bruhat intervals in Coxeter groups of type $A$ and apply this result to prove a conjecture of Brenti from 2003 concerning the computation of Kazhdan-Lusztig $R$-polynomials via special matchings. This yields new evidence in support of the Combinatorial Invariance Conjecture.

Multispecies spatial data arise in many applications where interactions between different entities are central to system behaviour, including biomedical imaging, geospatial analysis, and species ecology. Despite their importance, relatively few quantitative tools exist to capture such interactions. In this work, we propose magnitude-based features for the analysis of multispecies spatial data. Magnitude is a real-valued invariant of finite metric spaces that can be interpreted as an effective number of points, incorporating both spatial configuration and scale. We develop global and local magnitude feature vectors and demonstrate their utility on synthetic tumour microenvironment data, and in tissue microarray data from human colorectal cancer samples. Locally, the method identifies distinct neighbourhood types and reveals spatial heterogeneity; in the model, this includes radial patterns associated with different qualitative outcomes of the simulations, while in the real-world data it reflects the importance of tertiary lymphoid structure-like interactions between B and T cell populations. Globally, the approach recovers known classifications of long-term simulation outcomes across parameter regimes in synthetic data, and suggests important roles for CD4+ T cells and CD163+ macrophages in distinguishing patients with favourable Crohn's like reactions from unfavourable diffuse immune infiltration. Together, these results suggest that magnitude-based features provide a powerful and flexible tool for the analysis of multispecies spatial data.

Global nonnegative martingale solutions are shown to exist for a stochastic Keller-Segel system with a nonlocal Fisher-KPP source and super-linear multiplicative noise. The result is obtained for nonnegative initial data with no smallness assumption, provided that the nonlocal source term is dominant. The main difficulty stems from the absence of a coercive structure and the super-linear nature of the noise. An additional cut-off with finite L^2 norm in the classical Galerkin method is added to establish a well-posed approximation problem. Moreover, due to the nonlocal Fisher-KPP structure, it is necessary to prove the positivity of the approximating solution in order to obtain uniform estimates. In the compactness arguments, the usual tightness argument in the framework of Hilbert spaces cannot be directly applied to the uniform estimates obtained in this paper. As a result, we develop a more general version of the compactness argument and tightness criterion, presented in the appendix, which will be applied throughout the paper. This allows for the global existence of nonnegative martingale solutions to be derived from Jakubowski's version of the Skorokhod Theorem, along with a thorough discussion of the convergence properties.

Optimistic Gradient Descent Ascent (OGDA) and Optimistic Multiplicative-Weights Update (OMWU) are two very popular algorithms to solve convex/concave saddle-point problems, where OMWU is the non-Euclidean, entropic version of OGDA. It is known since the '80s that the last iterate of OGDA asymptotically converges to a saddle point in smooth problems. On the other hand, it is unknown if OMWU has the same property. In this paper, I show that OMWU converges asymptotically for smooth convex-concave saddle-point problems, with a small enough constant learning rate. The result does not require uniqueness, strict complementarity, an error bound, or initialization near a solution. The main new ingredient is a boundary argument showing that every cluster point satisfies the inactive-coordinate KKT inequalities. The boundary argument was discovered with assistance from ChatGPT and is documented in the appendix.

Originally motivated by creating first-person computer visualizations within Riemannian manifolds -- the author was led to study deformable-body mechanics, as rigid-body mechanics is not available in a generic Riemannian manifold due to its lack of nontrivial isometry group. Hyperelasticity is a particularly nice sub-category of continuum mechanics in which a deformable, elastic body's behavior is determined by a stored energy density function. This allows problems to be posed variationally, and powerful tools brought to bear on studying and solving them. This article presents numerical simulations of static solutions to a particular class of problems in hyperelastic mechanics in 2-dimensional Riemannian manifolds in which a flat hyperelastic body $B$ is embedded into a region $\Omega$ in a nowhere-flat surface $S$ of revolution $z=z\left(r\right)$ such that $\left|K\left(r\right)\right|$ decreases as $r\to\infty$, where $K$ denotes the Gaussian curvature of $S$. For example, the funnel $z=-r^{-1}$ or the paraboloid $z=\frac{1}{2}r^{2}$. Because $B$ is flat, the body can't achieve a zero-stored-energy configuration, and restorative forces arise in the body to move it toward a region of lower stored energy -- meaning, toward a flatter configuration. With the addition of a gravitational potential $U\left(r\right)=z\left(r\right)$ on $S$, forces act on the body to pull it toward $r=0$. If the body has sufficient stiffness and remains within the region $\Omega$, then the body has an equilibrium configuration in which the body's deformation-response forces perfectly cancel the gravitational forces. Such a configuration represents a kind of "levitation" phenomenon within this surface. The numerical implementation of this problem will be detailed and the resulting numerical solutions and various consequences discussed.

The process of deriving the Desarguesian plane $PG(2,q^2)$ to get the Hall plane is well known, and the problem of when a conic in $PG(2,q^2)$ inherits to an arc in the Hall plane has been solved. In this article we look at the generalisation of replacing an André net of $PG(2,q^t)$, $t\geq 3$ to construct an André plane of order $q^t$. This article looks at the case where $q$ is odd and $t$ is prime, and determines when a conic in $PG(2,q^t)$ inherits to an arc in an André plane. Further, the number of arcs in an André plane that are inherited in this way is enumerated.

We prove an exact Fourier-dimension formula for scalar dyadic Mandelbrot cascades pushed forward to fixed C^2 Jordan curves with nonvanishing curvature. Let W be in the minimal Kahane-Peyriere regime, let the scalar dyadic cascade live on T = R/Z, and let gamma map T to R^2 be a fixed C^2 Jordan curve with nonvanishing curvature, parametrized at constant speed. For the push-forward measure mu_gamma, we prove that, almost surely on non-extinction, its Fourier dimension is A_loc(W), the usual local exponent obtained by optimizing over q>1 from the moment expression involving E[W^q]. The upper bound follows from the scalar circle local-dimension theorem, bi-Lipschitz transfer to the fixed curve, and a deterministic curved-support obstruction for Fourier dimension. The lower bound follows from a fixed-curve finite-r annular theorem, which gives summable annular Fourier decay under a single finite moment witness. The main analytic input is a deterministic phase-geometry package for fixed nondegenerate C^2 curves: stationary tubes, derivative bands, and phase-bin coefficient estimates replacing the explicit trigonometric structure available on the unit circle.

A graph $G$ is $k$-$colorable$ if $V(G)$ can be partitioned into at most $k$ stable sets. A graph $G$ is $k$-$chromatic$ if $k$ is the smallest integer for which $G$ is $k$-colorable. In general, for a fixed $k\ge 3$, determining whether an arbitrary graph $G$ is $k$-colorable is NP-complete. Consequently, $k$-coloring algorithms for restricted graph classes, such as $\mathcal{H}$-free graphs, have been widely studied over the past few decades. A graph $G$ is $k$-$vertex$-$critical$ if $G$ is $k$-chromatic and every proper induced subgraph of $G$ is ($k$-1)-colorable. Given a graph $G$, most of the certifying $k$-coloring algorithms in the literature either output a $k$-coloring of $G$ or a ($k$+1)-vertex-critical induced subgraph of $G$, thus, proving that $G$ is not $k$-colorable. As a result, $k$-vertex-critical graphs have gathered considerable attention in the recent years. Beaton and Cameron [Vertex-critical graphs in co-gem-free graphs, Theoretical Computer Science 1042 (2025) 115234] asked for which graphs $H$ of order five are there finitely many $k$-vertex-critical (co-gem, $H$)-free graphs for all $k$? In this paper we explore the structure of (co-gem, house)-free graphs and (co-gem, dart)-free graphs, and prove that, for each $k\ge 1$, there are finitely many $k$-vertex-critical (co-gem, $H$)-free graphs, when $H$ is in $\{$house, dart$\}$.

We study the full automorphism groups of smooth cubic fourfolds with prescribed symplectic automorphism group. Our starting point is the classification of symplectic automorphism groups by Laza and Zheng. We focus on the non-symplectic index, namely, the index of the symplectic automorphism group in the full automorphism group. We prove general restrictions on this index. We also compute bounds by group-theoretic and lattice-theoretic methods. In several cases, we determine all possible indices. For coinvariant lattices of rank 19, we classify all possible pairs consisting of the symplectic automorphism group and the full automorphism group.

In 'On Denoting' Russell proposed the most influential theory of definite descriptions, expressions of the form 'the F'. Characteristic for Russell's approach is that definite descriptions are not treated as what they appear to be on the surface, i.e. as singular terms. Instead they are eliminated by a contextual definition. Russell formalises definite descriptions in the context of complete sentences of the form 'The F is G'. This requires scope markers to distinguish, e.g., internal from external negation. It was recognised by Burge, and Kalish and Montague, however, that the essential features of Russell's approach may be formalised while respecting the syntactic category to which definite descriptions appear to belong. An alternative, favoured by Neale, follows Russell in that complete sentences 'The F is G' are formalised by a binary quantifier. The undeniable importance of the theory of definite descriptions for logic, mathematics and philosophy demands that it be formalised to meet the standards of modern proof theory. This is the topic of the present paper. We systematise, compare and extend existing approaches. After presenting its essential features, we formalise Russell's theory of definite descriptions in sequent calculus. Three approaches will be considered. The first uses a binary quantifier, whereas the remaining two employ the term-forming iota operator. The first of these employs only the iota operator, the other employs in addition the lambda operator which does duty as a scope marker. All systems satisfy the standards for modern proof theory, in particular cut elimination. The appendix reformulates these systems in natural deduction, which is more convenient for practical purposes.

This paper proves normalisation theorems for intuitionist and classical positive free logic, without and with the iota operator for definite descriptions `the F'. Positive free logic also opens a number of options for rules for iota. In total, six different formalisations of theories of definite descriptions will be discussed, three proposed by Lambert, and three alternatives. The latter are motivated by considerations relating to proof-theoretic harmony between introduction and elimination rules. The philosophical importance of the various systems and results is indicated. The paper builds on Kürbis (2025), but is largely self-contained. The proofs for the present systems are easier than those for negative free logic.

This paper presents a novel class of high-order mass-, energy- and momentum-preserving exponential integrators for solving the derivative nonlinear Schrödinger equation. Firstly, we reformulate the original system into an exponential supplementary variable system based on the idea of the exponential supplementary variable approach, and then the reformulated system is discretized by using the standard Fourier pseudo-spectral method in space and the high-order prediction and correction Lawson Runge-Kutta method in time, respectively. The proposed method is highly efficient, temporally high-order accurate, and simultaneously preserves the mass, energy and momentum in the discrete setting. Finally, numerical experiments validate the accuracy and energy-preserving properties.

Let $X$ be the Segre product of three projective lines. For a fixed effective divisor $D$ on $X$, we introduce the notions of $D$-'t Hooft, $(D_i,D_j)$-special and $D$-sectional special bundle. The varieties parameterizing these bundles yield a natural stratification of the moduli space of stable instanton bundles with fixed Chern classes. After characterizing the curves associated with these bundles via Serre correspondence, we describe the corresponding Hilbert schemes. Using this description, we analyze the moduli spaces of $h_i$-'t Hooft bundles and the smaller strata of $(h_i,h_j)$-special and $(h_i)$-sectional special bundles. Finally, we provide a detailed study of the low-charge cases.

This document is a pdf generated from old plain-TeX files of 1986, of Nils Lid Hjort's `Notes on the Theory of Statistical Symbol Recogntion', a limited circulation 207-pages monograph published at the Norwegian Computing Centre, as Report no. 778/1986. It gives the basics of the statistical pattern recognition theory developed to suit the needs of several industrial projects, related to contracts with the Norwegian-German firm SysScan, the Royal Norwegian Research Council, and yet others, involving symbol recognition and classification analysis from noisy images, related to maps, documents, satellite imaging, etc. The methods and algorithms developed also needed to fit the technology of that time, anno c. 1986, with machines scanning documents, converting these to vector representation, within computational and machine system boundaries. There is an accompanying and also limited circulation booklet, `Statistical Symbol Recognition: Development of a System', by Knut Bråten, Erik Holbæk-Hanssen, and Torfinn Taxt (Report No.777/1986, Norwegian Computing Centre, Oslo), detailing the system developments. Thus developments took form and shape on two frontiers, in close collaboration, Hjort's statistical methods and getting the technology to work, with its multiple components, hardware and software.

We show that if $M$ is a metric space of Nagata dimension at most $d$, then there exists an atlas on $M$ modeled on $\mathbb R^d$ such that every Lipschitz map $f:M\to Y$ (with values in an arbitrary Banach space $Y$) can be uniformly approximated by maps that are affine, and thus $\mathcal{C}^1$-smooth, with respect to this atlas. The construction relies on random metric partitions and stochastic retractions inside Lipschitz-free spaces. As an application, we introduce approximate continuous upper gradient $X$-structures (ACUG $X$-structures) on metric spaces and prove that every space of finite Nagata dimension carries an ACUG structure modeled on a superreflexive Banach space. Finally, adapting a proof due to Bourgain, we show that if $M$ has an ACUG superreflexive-structure, then the Lipschitz-free space $\mathcal{F}(M)$ has Pelczyński's property (V*). In particular, at least in the compact case, our result recovers all previously known examples of metric spaces $M$ for which $\mathcal{F}(M)$ has property (V*).

We formulate and prove a conditional finite-scale reduction theorem for the local one-component regularity problem for suitable weak solutions of the three-dimensional Navier--Stokes equations. Starting from a scale-invariant bound Phi(1) <= M and smallness of the critical vertical component C_3(1) = delta, the argument compares the solution with a strict two-and-a-half-dimensional shadow class. The comparison is made in the harmonic-pressure quotient, which is the natural local topology for pressure compactness. The Reynolds commutator produced by coarse graining is treated as a positive covariance stress and is absorbed by an unresolved-variance buffer; consequently this stress contributes additively, while the genuinely vertical residuals carry a positive power of delta and may pass through finite-stage exponential constants. The theorem is deliberately stated as a reduction theorem. Under the explicitly listed structural inputs--prepared pressure-covariance closure, weak horizontal-defect admissibility, sharp admissible-time trace tightness, singular-stratum tangent-cone inputs, strict limiting smoothing and decay, finite-window trace-cost/Newton solvability, and the vertical-duality active-residual estimate--we derive r_reg(0,0) >= c_{M,theta} |log C_3(1)|^{-sigma/3}. The paper does not constitute an unconditional resolution of the logarithmic one-component regularity problem. Its contribution is a theorem-driven reduction: strict-shadow selection failure is reduced to a finite-mode flat trace obstruction, and that obstruction is eliminated, conditionally, by vertical duality forced by the full three-dimensional vertical momentum equation.

The classical p-adic Cauchy theorem asserts that formal solutions of ordinary p-adic differential equations are convergent. In this article we establish a geometric analogue of this result for arbitrary smooth rigid-analytic varieties. More precisely, we show that the horizontal sections functor defined using the overconvergent de Rham period structure sheaf agrees with Scholze's horizontal sections functor defined using OBdR. Equivalently, every formal solution arising from Scholze's construction is already overconvergent. As an application, we identify Scholze's horizontal sections functor with the de Rham functor for D-cap-modules on vector bundles with flat connection.

In this article, we present a new approach to studying multivariate period rings that is more consistent with classical theory and provides a clearer description of their structure. We also prove that the category of $B$-admissible representations forms a Tannakian subcategory of the category of representations of $G_{K,\Delta}$ by defining an analogue of $(F,G)$-regular rings, which is central to the classification of representations in multivariate $p$-adic Hodge theory.

In this paper, we investigate finite and infinite decomposition formulas for zero mean curvature (ZMC) graphs in three-dimensional spaces with different metrics, including Euclidean space, Lorentz--Minkowski space, and isotropic 3-space. We first derive new Euler--Ramanujan-type identities yielding decompositions for the conjugate of Scherk's first minimal surface in terms of dilated catenoids. We then employ Weierstrass factorisation and power series techniques to obtain infinite decompositions for a broad class of ZMC graphs in isotropic 3-space into helicoids, logarithmoids of revolution, and Enneper surfaces. We further extend these decomposition results to larger families of ZMC surfaces across these spaces by considering surfaces arising from the López--Ross transformation, Bonnet rotation, and a one-parameter family of metric deformations. We also investigate finite decompositions, establishing finite analogues of Scherk tower decompositions in both Euclidean and isotropic settings. In addition, we prove a theorem characterising finite decompositions of isotropic minimal surfaces. Finally, we discuss applications of the resulting decomposition theory to lamellar structures.

Let $E$ be a flat Lorentzian space of signature $(2,1)$. A Margulis space-time is a noncompact complete flat Lorentzian $3$-manifold $E/\Gamma$, where the holonomy group $\Gamma$ is a free group of rank $g\geq 2$ acting freely and properly discontinuously by isometries. We consider the case where $\Gamma$ contains a parabolic element. We show that sufficiently small deformations of $\Gamma$ still act properly discontinuously on $E$ provided their linear parts are Fuchsian; moreover, the number of conjugacy classes of parabolic elements may increase or decrease under deformation. Our proof combines our previous compactification of $E/\Gamma$ relative to parabolic holonomy elements with a partial generalization of the work of Carrière. However, this result depends only on the parts on parabolic actions of our earlier work. We believe that the shortness of the proof of this openness result is of independent interest.

Let $A$ be a finite-dimensional basic algebra over an algebraically closed field $K$, $T$ a finitely generated $\tau$-tilting right $A$-module and $B={\rm End}_A T$. Denote by ${\rm Fac}T$ the subcategory of finitely generated right $A$-modules generated by $T$. We define the depth relative to $T$ and the delooping level relative to $T$ and show that the finitistic dimension of the opposite algebra of $B$ is bounded by the depth of $\textup{Fac}T$ relative to $T$ and the delooping level of $\textup{Fac}T$ relative to $T$. We give applications to the finitistic dimension conjecture. More precisely, we show that if $A$ is a minimal representation infinite algebra or an algebra of finite representation type, then the finitistic dimension of $B^{op}$ is finite.

We study Schrödinger operators with a one-center point interaction, formally defined by \begin{align*} -\Delta_\alpha=-\Delta+\alpha\,\delta_0(\cdot), \end{align*} for $\alpha\in\mathbb{R}$, and the associated heat equation \begin{align} \partial_t u=\tfrac{1}{2}\Delta_{\alpha} u,\quad u(0,x)=u_0(x)\in C_c^{\infty}(\mathbb{R}^3\setminus\{0\}).\label{eq:HEapp} \end{align} Here $\Delta$ denotes the Laplacian (self-adjoint on $L^2(\mathbb{R}^3)$) and $\delta_x$ the Dirac measure at $x$. The operator $-\Delta_\alpha$ can be realized either as a self-adjoint extension of $-\Delta|_{C_0^{\infty}(\mathbb{R}^3\setminus\{0\})}$ in $L^2(\mathbb{R}^3)$, or as the norm-resolvent limit of $-\Delta+\lambda_\varepsilon V(\cdot/\varepsilon)$ for suitable $\lambda_\varepsilon$ and $V:\mathbb{R}^3\to\mathbb{R}$. In this paper we construct, for each $t>0$ and $x\in\mathbb{R}^3\setminus\{0\}$, a probability law on path space and a normalizing function $G_t^\alpha(x)$ giving the following probabilistic representation of the solution to the associated equation: \begin{align*} u(t,x)=G_t^\alpha(x)\,\mathbb{E}\bigl[u_0\bigl(W^{t,x}(t)\bigr)\bigr], \end{align*} where $\{W^{t,x}(s):0\le s\le t\}$ is a continuous process depending on $(t,x,\alpha)$. The result provides a Feynman--Kac type formula for the heat equation with a one-point interaction in three dimensions.

We consider the homogenization of three dimensional viscous incompressible non-Newtonian flows satisfying certain general $r$-structure in perforated domains. We focus on the case of `small holes' by assuming the holes under consideration are of size $\varepsilon^{\alpha}$ with $\alpha >3$, where $\varepsilon$ is the perforation parameter used to measure the mutual distance between the holes. We show the limit equations remain unchanged in the homogenization limit under the constraint $ \frac{6(\alpha - 1)}{4\alpha-5}< r<3-\frac{3}{\alpha}$, which seems optimal in the sense of Sobolev capacity of holes as explained in Remark 1.3. Quantitative convergence rates are further derived for both the velocity field and the pressure. To the best of our knowledge, both the qualitative and quantitative homogenization results are firstly given for non-Newtonian flows in the `small holes' case.

We study discrete analogues of classical spectral geometric inequalities and extremal eigenvalue problems on graphs. The well-known Krahn--Szegő inequality states that the minimum of $\lambda_2(\Omega)$ among bounded open sets of $\mathbb{R}^n$ with given volume is achieved by the union of two identical balls $\mathbb{R}^n$. Firstly, we establish a Krahn--Szegő type inequality for trees. For trees with a fixed number of interior vertices and boundary leaves, we completely characterize the extremal structures that minimize the second Dirichlet eigenvalue. Secondly, we develop a nodal domain method for adjacency matrices. By proving a nodal domain theorem in adjacency version for graphs, we obtain upper bounds for the second largest adjacency eigenvalue $\rho_2(G)$ of $G$ in given graph classes. These bounds imply some previous results. Finally, we settle the Aouchiche--Hansen conjecture (2010) on the second largest eigenvalue with given number of edges and clique number. We prove that for connected graphs $G$ of odd order $n \geq 5$, $|\rho_2| \cdot \omega \leq m-2$, with equality if and only if $G$ consists of two complete graphs of orders $\frac{n+1}{2}$ and $\frac{n-1}{2}$ joined by an edge or a path. For even $n \geq 2$, the quantity $|\rho_2| \cdot \omega - m$ is maximized exactly when $G$ is the join of two copies of $K_{n/2}$ by an edge. The core of the methods developed in this paper is to regard a connected graph as an internally disconnected graph with Dirichlet boundary condition. This perspective allows us to transfer nodal domain techniques from continuous spectral geometry to discrete settings and to obtain sharp extremal characterizations across diverse graph classes.