Papers

Let $\mathcal K$ be the convex hull of a planar Brownian motion up to time $1$. We explicitly determine the probability that $\mathcal K$ contains a prescribed point $z\in\mathbb R^2$. The proof is based on a general formula relating the radial function of a random convex body to its support function and its derivative, together with a comparison of $\mathcal K$ with an arcsine-rescaled Gaussian ellipse. We also obtain an analogous formula for the convex hull of a planar Brownian bridge. Finally, we derive small- and large-time asymptotics for the corresponding coverage probabilities.

We study the first exit of an isotropic planar Ornstein-Uhlenbeck process from an annular sector, the region bounded by two concentric circular arcs and two radial segments, and obtain explicit eigenfunction expansions for the associated exit functionals. After the ground-state transformation that renders the generator self-adjoint, polar separation reduces the problem to a radial confluent hypergeometric equation whose two independent (Whittaker) solutions of generally non-integral order are combined through a two-radius determinant that fixes the spectrum. In this way we obtain the survival probability, the density and moments of the exit time, and -the principal contribution -the joint law of the exit time and the exit boundary, which resolves both the instant of exit and the boundary piece through which it occurs. As a companion separable case, we also treat a genuinely correlated, reversible planar Ornstein-Uhlenbeck process on a principal-axis rectangle, where the radial functions are replaced by parabolic cylinder functions: the survival probability factorizes into one-dimensional problems, while the joint law of the exit time and the exit side does not. The expansions require only one-dimensional root-finding and quadrature and are validated against Monte Carlo simulation. An application to an optically trapped colloidal particle is discussed, a setting in which the annular geometry arises naturally.

We study a representative-agent Epstein-Zin economy with geometric dividends and a hidden finite-state Markov drift. We allow the price-dividend ratio to contain an additional positive, absolutely continuous valuation factor and, within the class $\mathfrak C$ defined below and under the regularity, admissibility, and positivity conditions of our main theorem, equilibrium forces this factor to be constant, yielding belief-Markovian prices. In the two-state case, under the stated positivity condition and strictly positive transition intensities, we prove existence, uniqueness, endpoint smoothness, interior analyticity, and uniform bounds for the positive solution of the pricing equation. For $0<\theta\leq1$, this solution supports an equilibrium under any continuous short rate satisfying the model's one-sided portfolio condition. Finally, in the two-state subregion $\eta>0$, we derive belief-dependent stock volatility, a European option-pricing PDE, and a leading short-maturity conditional risk-neutral log-return skewness expansion.

Let $G$ be a simple graph of even order at least four, and let $K_G$ denote the complete graph on $V(G)$. A perfect matching of $K_G$ is called a pairing of $G$. The graph $G$ has the Pairing-Hamiltonian property, or PH-property, if every pairing $M$ of $G$ admits a perfect matching $N\subseteq E(G)$, disjoint from $M$, such that $M\cup N$ is a Hamiltonian cycle of $K_G$. We prove that the PH-property is preserved under Cartesian products. More precisely, for graphs $G$ and $H$ of even order at least four, we show that both $G$ and $H$ are PH if and only if every pairing of $G\square H$ admits a Hamiltonian completion contained in a spanning union of vertex-disjoint prisms determined by a perfect matching of $G$ or of $H$. Without this support restriction, the converse fails: a Cartesian product may be PH even when neither of the two graphs is PH.

We prove that every bounded stable entire solution $v$ of the Allen–Cahn equation in $\R^3$ is one-dimensional. As a consequence, the full De Giorgi conjecture in $\mathbb{R}^4$ is true. We also obtain local curvature estimates for stable solutions. The proof strategy is inspired by the recent breakthrough work of Chan, Fernández-Real, Figalli and Serra [J. Amer. Math. Soc. 2026], by reducing the stabilty condition for the Allen-Cahn equation to a weak stability condition on a surface (the zero set) and then utilizing Gauss-Bonnet formula. For this purpose, we first use the stability condition to get a sublinear bound for a weighted integral that controls the zeros where the solution is far from planar. If such zeros exist, we isolate a bounded set of them and join $1-v^2$ near this set to derivatives of one-dimensional transitions farther away. By controlling the interaction between these transitions, we derive the weak stability condition on the zero set, which is then used to bound a weighted integral of the squared curvature on the regular part of the zero set by a cutoff gradient integral and a controlled error. We use this inequality to bound the intrinsic area and construct logarithmic cutoffs. The resulting compactly supported test function has a negative contribution near this set that exceeds all joining and cutoff errors, contradicting stability.

We determine the full automorphism group of every connected Cayley graph on an almost simple group with a normal connection set. We also characterize exactly when the full automorphism group is generated by right translations, group automorphisms preserving the connection set, and inversion. Our results substantially generalize several results in the literature, including the known determinations of the automorphism groups of derangement graphs, complete transposition graphs and complete alternating group graphs. As a further application, we establish criteria for a finite nonabelian simple group to admit a graphical doubly regular representation and determine exactly which alternating groups do so, correcting claims in the literature.

In this paper, we develop a polyhedral version of Milman's centro-affine spectral framework for full-dimensional origin-symmetric polytopes. A weighted graph Laplacian on the facets plays the role of the smooth centro-affine Laplacian. For $n\geq 3$, we prove that its first nonconstant even eigenvalue satisfies \[ \lambda_{1,e}(P)\geq n-1+\frac{1}{n-1} \] whenever $P$ is a local minimizer of $\lambda_{1,e}$ among origin-symmetric polytopes with the same facet normals, but not necessarily the same normal fan.

The main purpose of this paper is to introduce and study restricted Lie color triple systems. We consider first restricted Lie color algebras and then provide a definition and relevant properties of restricted Lie color triple systems. Moreover, we introduce and discuss representations of restricted Lie color algebras and restricted Lie color triple systems.

Let $\mu$ be a positive Borel measure on $[0,1)$ and $\omega$ a radial weight. In this paper we consider the Cesàro-type operator $C_{\mu,\omega}$ induced by the reproducing kernel $B^{\omega}$ of the weighted Bergman space $A^2_{\omega}$, given by $$ C_{\mu,\omega}(f)(z)=\int_{0}^{1}f(tz)B^{\omega}_t(z)\,d\mu(t), \quad z \in \mathbb{D}, $$ for functions $f$ analytic in $\mathbb{D}$. Under the assumption that $\omega$ satisfies a natural doubling property, we study the boundedness of $C_{\mu,\omega}$ acting on several spaces of analytic functions, including Hardy spaces $H^p$ and weighted Bergman spaces $A^p_{\nu}$. For $0<p,q<\infty$ and a two-sided doubling weight $\nu$, we completely characterize when $C_{\mu,\omega}: H^p \to H^q$ and $C_{\mu,\omega}: A^p_{\nu} \to A^q_{\nu}$ are bounded in terms of the interplay of tail integrals or moments of the inducing weights and the measure $\mu$. Many of the results obtained are new even in the setting of standard weights or when the Bergman reproducing kernel is replaced by the Cauchy kernel. In addition, we consider $C_{\mu,\omega}$ acting on $H^{\infty}$, Korenblum spaces and weighted Hardy spaces.

Symmetry in the context of equivalence or isomorphism is a fundamental and natural concept in any study of discrete structures. Symmetries are also important for non-discrete structures, but their treatment can be more challenging and is perhaps therefore often overlooked. This holds for many studies of complex Hadamard matrices, that is, matrices with unimodular complex entries satisfying the equation $HH^{\dagger} = nI$, where $H^{\dagger}$ is the conjugate transpose of $H$. In the current work, equivalence of complex Hadamard matrices is considered, and algorithms for determining equivalence of matrices and the automorphism group of a matrix are presented. The algorithms are used to establish the automorphism group of a large number of complex Hadamard matrices from the literature.

Let $\sigma\mathrm{Mex}(n)$ and $\sigma_e\mathrm{Mex}(n)$ denote the sum of minimal excludants and sum of even minimal excludants over all overpartitions, respectively. In this work, we study these two combinatorial objects from an arithmetic perspective. We prove that, for $n\geq 0$, $\sigma_e\mathrm{Mex}(n)\equiv\sigma\mathrm{Mex}(n)-\bar{p}_{\geq 2}(n)\pmod{2^2}$, where $\bar{p}_{\geq 2}(n)$ denotes the number of overpartitions of $n$ with all parts at least $2$, and obtain relation with basic hypergeometric series. Furthermore, we prove the asymptotic behavior of $\sigma\mathrm{Mex}(n)$, $\sigma_e\mathrm{Mex}(n)$, and $\bar{p}_{\geq 2}(n)$ as $n\to\infty$. In particular, we prove that $\sigma\mathrm{Mex}(n)\sim 2\,\sigma_e\mathrm{Mex}(n).$

A family of permutations in $S_n$ is called even-intersecting if every two distinct members agree in an even number of positions. Let $M(n)$ denote the maximum size of such a family. For even $n$, we prove that $$n!!\leq M(n)\leq e^{\frac{n}{2}+o(n)}n!!,$$ improving the bound obtained from a theorem of Cameron, Deza and Frankl (1987) by an exponential factor. This problem may be viewed as a permutation analogue of the classical Eventown problem for set systems. For odd $n$, we give a construction yielding $M(n)\geq n^2/4$. We further extend this construction to obtain $$ M(n)\geq\bigl(n-\sqrt{n-1}\bigr)^2 ,$$ whenever $n=(q+1)^2+1$ and $q$ is an odd prime power. The latter bound asymptotically matches the upper bound $M(n)\leq(n-1)^2+1$ obtained by Cameron, Deza and Frankl.

We develop a categorical framework for Lie-Rinehart monoids and their weak modules in a symmetric monoidal category. Using crossed homomorphisms, we construct a natural action of the monoidal category of modules over a Lie monoid on the category of weak Lie-Rinehart modules, thereby obtaining categorical versions of the Shen-Larsson functors. We further characterize the conditions under which the category of weak modules admits a monoidal structure and identify the corresponding condition for the associated functors to be strict monoidal. A dual theory for Lie- Rinehart comonoids and weak comodules is developed using cocrossed homomorphisms. Combining the module and comodule constructions, we obtain a bimodule category structure on the category of weak modules. Finally, we specialize the general framework to the symmetric monoidal category of super vector spaces, recovering Lie-Rinehart superalgebras and their associated Shen-Larsson-type constructions.

We show that for every smooth complex cubic threefold $Y$, the fourfold $Y\times\mathbb{P}^1$ is irrational. As an application, we construct a smooth stably rational complex Fano variety $X$ which is irrational, while $X^m$ is rational for every $m\geq2$ and $\mathrm{Sym}^2 X$ is rational.

In this paper, we present a refinement of the symplectic Lidskii theorem for directional derivatives of symplectic eigenvalues, which is inspired by the work of Sendov [Electron. J. Linear Algebra 41(2025), 338-341]. We show that for a $2n \times 2n$ real positive definite matrix $A$ and symmetric matrices $H, K$ \begin{align*} d'(A;H+K)-d'(A;H) \prec_{\pi_A} d'(A;K). \end{align*} Here $\prec_{\pi_A}$ represents block-majorization corresponding to the partition $\pi_A$ of $\{1,\ldots, n\}$ given by the equality blocks of symplectic eigenvalues of $A$. We know that Hermitianization of a symmetric matrix occurs in the directional derivative expression of symplectic eigenvalues. We establish a componentwise inequality comparing the symplectic eigenvalues of a positive definite matrix with the ordinary eigenvalues of its associated Hermitianization, and also characterize the equality case. By using the aforementioned inequality, we then show that the symplectic Lidskii theorem follows from the directional derivative Lidskii theorem.

The notion of induced packing treewidth aims to unify classes defined by forbidden induced subgraphs or induced minors with classes defined by the existence of certain structured tree decompositions. For a graph $H$, \emph{induced $H$-packing treewidth}, denoted by $\treepi_{H}$, is a tree-decomposition-based graph parameter that, for each bag, measures the maximum number of pairwise anticomplete induced copies of $H$ intersecting that bag. This notion generalizes some previously studied parameters: when $H=P_1$, it is equivalent to tree-independence number, and when $H=P_2$, it is equivalent to induced matching treewidth. We prove the following: \begin{itemize}[itemsep=2mm,leftmargin=6mm] \item For all $a,t\in \mathbb{N}$, $K_{a,a}$-free graphs of bounded induced $P_t$-packing treewidth have bounded tree-independence number. This extends the previous result of Abrishami et al. [SIAM J. Discrete Math., 2025] for $t=2$, and a result of Hajebi and Spirkl who showed that $(P_t,K_{a,a})$-free graphs have bounded tree-independence number. \item If $H$ is any fixed path or a star, then the class of graphs of bounded induced $H$-packing treewidth is $\chi$-bounded. Again, this extends the previous result of Abrishami et al. [SIAM J. Discrete Math., 2025] for $H=P_2$. \item Finally, we study the relationship between induced packing treewidth and sim-width, a width parameter based on branch decompositions. We show that, although sim-width and induced $P_3$-packing treewidth are incomparable, graphs of bounded sim-width that exclude all \emph{$H$-obstructions}—certain graphs that force large induced $H$-packing treewidth—have bounded induced $H$-packing treewidth. This simultaneously generalizes and resolves questions posed by Abrishami et al. [SIAM J. Discrete Math., 2025] and Brettell et al. [European J. Comb., 2025]. \end{itemize}

We prove that no family of two continuous self-maps witnesses that the circle is a topological fractal, answering a question of Karasová and the present author. Since three maps are known to suffice, this bound is optimal. In contrast, for every $\varepsilon>0$ there are two continuous self-maps of the circle, depending on $\varepsilon$, whose images cover the circle and an integer $N$ such that every composition of $N$ of them has image of diameter less than $\varepsilon$. Thus two maps suffice at any prescribed scale, but no fixed pair works at all scales.

In this paper, we construct a Gelfand model for the Okada algebra $O_n(X,Y)$ with generic parameters $X$ and $Y$, on the space of symmetric Okada arc diagrams using a conjugation-type action. The model is constructed inductively by identifying the Okada algebra as a diagram algebra and using the Jones basic construction to obtain a tower of algebras that are themselves Okada algebras at lower levels. We use the model to obtain all the irreducible representations of $O_n(X,Y)$, indexed by the elements of rank $n$ of the Young–Fibonacci lattice, and identify them with the cell modules of $O_n(X,Y)$.

Let $\mathcal A$ be a complete deterministic finite automaton on a state set $Q$ of size $n$ with $k$ letters, and for a proper nonempty subset $S$ of $Q$ let $\mathrm{minext}(S)$ be the length of a shortest word $u$ with $|Su^{-1}|>|S|$, where $Su^{-1}=\{q: q\cdot u\in S\}$. To each state $q$ attach the integer $\beta^{\ast}_q=\sum_{t=1}^{n-1}k^{\,n-1-t}(\mathrm{indeg}_t(q)-k^{t})$, where $\mathrm{indeg}_t(q)$ counts the pairs $(p,u)$ with $|u|=t$ and $p\cdot u=q$, and let $B(S)=\sum_{q\in S}\beta^{\ast}_q$. On every synchronizing automaton, $B(S)\ge0$ implies $\mathrm{minext}(S)\le n-1$, so, as $B(Q)=0$, one of $S$ and $Q\setminus S$ extends within $n-1$; when $B(S)>0$ no hypothesis is needed. Kari's Eulerian extension lemma is the case $\beta^{\ast}=0$, and $\beta^{\ast}$, like every member of the family $\sum_{t=1}^{n-1}c_t\sigma_t$, $c_t>0$, vanishes identically if and only if the automaton is Eulerian, where $\sigma_t(S)=\sum_{q\in S}(\mathrm{indeg}_t(q)-k^{t})$. On strongly connected automata $\sigma_t(S)/k^{t}$ has Cesàro limit $n\,e(S)/e(Q)-|S|$ for Friedman's weight $e$; that limit certifies singletons but no larger subset in general. The hypothesis $B(S)\ge0$ cannot be relaxed by one integer unit, nor can the constant $n-1$ be improved. A second-moment test on the sizes $|Su^{-1}|$ certifies 60 to 95 percent of the subsets with $B(S)<0$ at $n\le7$. Along non-Eulerian automata whose words of length $n-1$ merge a fraction of the state pairs bounded below, with $\max_q\mathrm{indeg}_{n-1}(q)=o(nk^{n-1})$, it certifies all but a vanishing share of them. The functional $B$ certifies half of the subsets outside $\{B=0\}$. At each subset size coprime to $n$ ($n\ge4$) some synchronizing Eulerian binary automaton attains the constant $n-1$; whether only there is open. No reset bound follows: Černý's automata have subsets not extending within $n-1$.

Given a cyclic code $C_g = \langle g(x) \rangle$ of odd length $m$ over $\mathbb{Z}_4$, one common way to build a quasi-cyclic (QC) code is to pick $f_1, \dots, f_\ell \in \mathbb{Z}_4[x]$ and let $C = \langle (f_1 g, \dots, f_\ell g) \rangle$. Because $\mathbb{Z}_4$ is not a field, the type of $C_g$ ($4^{k_1}2^{k_2}$) is not necessarily inherited by $C$. Determining conditions under which the type of $C_g$ is inherited by $C$ was posed as an open problem recently in \cite{AydinLuOnta2023}. In this paper, we settle this problem. We first give two sufficient conditions for the type to be preserved: one requires a single $f_i$ to be coprime to $x^m-1$ over $\mathbb{F}_2$, the other only requires the $f_i$ to be jointly coprime to it. Neither condition is necessary in general. Using the fact that $x^m-1$ is squarefree for odd $m$, we decompose $\mathbb{Z}_4[x]/\langle x^m-1\rangle$ into a product of finite chain rings using Chinese remainder theorem (CRT) and derive a condition on $f_1, \dots, f_\ell$ that is both necessary and sufficient for $C$ to match the type of $C_g$. This condition depends only on the irreducible factors of $x^m-1$ where $g$ does not already vanish. This also yields a simple test for when $C$ is a free $\mathbb{Z}_4$-module. Finally, we report many new QC codes over $\mathbb{Z}_4$, found by computer searches using Magma software~\cite{Magma1997} guided by this criterion, with Lee distances greater than previously known codes of the same type.

The kissing number in dimension $d$ is the largest number of non-overlapping congruent spheres that can simultaneously touch a central sphere of the same size. We study dimensions $25$-$31$, where the best previous constructions are based on Leech lifting from the optimal kissing configuration in dimension $24$. Our method exploits the absence of contacts between the unlifted bulk and the block consisting of lifted and auxiliary vectors. Rotating this block while keeping the bulk fixed creates room for two antipodal points in dimensions $26$, $27$, and $28$, and one point in dimension $29$. Three further modifications yield improvements in dimensions $25$, $30$ and $31$: (a) a nonorthogonal diagonal linear deformation of the lifted block admits two antipodal points in dimension $25$; (b) rotating the additional coordinates of the lifted vectors and then applying a small orthogonal transformation to the resulting lifted block as a whole admits two antipodal points in dimension $30$; (c) rotating only the additional coordinates of the lifted vectors admits four nonantipodal points in dimension $31$. Together, these constructions yield the new lower bounds $\tau_{25}\geq 197058$, $\tau_{26}\geq 198552$, $\tau_{27}\geq 200046$, $\tau_{28}\geq 204522$, $\tau_{29}\geq 209497$, $\tau_{30}\ge 220442$, and $\tau_{31}\geq 238354$.

We construct Riesz sequences whose Beurling density is arbitrary close to the critical density in the setting of reproducing kernel Hilbert spaces on metric measure spaces. In particular, our results apply to exponential systems on unbounded sets, nonlocalized Gabor systems and general coherent systems arising from nilpotent Lie groups. The methods used in this paper are based on our previous work on the redundancy of frames and a selector form of Weaver's conjecture.

Fully-actuated multirotor aerial vehicles must not only track nominal wrenches but retain the "readiness" to modulate them rapidly under disturbances. Classical effort-minimizing allocators ignore this dynamic limit, whereas maximizing readiness leads to topologically disconnected optimal sheets demanding physically impossible actuator rates. Enforcing a readiness safety floor on fixed-geometry symmetric platforms further encounters a zero-sum degeneracy: motor-speed redistribution cannot improve authority without conceding wrench tracking. This paper uses active morphology to break the degeneracy, treating servo tilt as a geometric resource supplying authority-recovery directions unavailable to static rotors. We construct a configuration-dependent, motor-only readiness certificate - the log-volume of the reachable wrench-rate set - that explicitly excludes servo capacity, preventing a "ghost capacity fallacy" in which the certificate would falsely credit slow mechanical kinematic limits instead of collapsing accurately at motor saturation. The certificate is enforced as a Control Barrier Function (CBF) within a Unified Physical-Command Quadratic Program acting on motor torques and servo setpoints. Closed-loop simulations of an articulated octorotor under severe gust disturbances show classical allocators diverging and uncertified articulated allocators violating the safety floor, while the proposed CBF filter bounds the system state and preserves vehicle authority.

We determine, up to isometry, the Riemannian tangent and normal cones at arbitrary faces of boxes, acute orthocentric simplices, rectangular simplices, and asymmetric crosspolytopes in spaces of constant sectional curvature $\kappa\in\mathbb R$. Using these descriptions, we derive explicit one-dimensional integral formulas for their internal and external angles in arbitrary dimension. In particular, our results cover regular simplices, cubes, and crosspolytopes, the three families of regular polytopes existing in every dimension. We also establish a correspondence between regular simplices and cubes with isometric local cones at corresponding faces, provided their edge-length parameters are suitably matched. Combining our angle formulas with the Poincaré relation, we obtain volume formulas for simplices and crosspolytopes in even dimensions. For boxes, we derive volume formulas valid in every dimension. Among the new results are angle formulas for spherical and hyperbolic cubes and crosspolytopes, volume formulas for asymmetric crosspolytopes, and a one-dimensional integral representation for the internal angles of regular crosspolytopes that appears to be new even in the Euclidean case. The formulas involve the standard normal distribution function $\Phi(x)$ and its values at purely imaginary arguments $\Phi(\mathrm{i} x)$. For example, if $d\geq 2$ is even, then the volume of the ideal regular simplex in the $d$-dimensional hyperbolic space of curvature $\kappa=-1$ is given by the absolutely convergent integral $$ \frac{\pi^{d/2}} {\sqrt{2}\,\mathrm{i}^{d}\, \Gamma\left(\frac{d+1}{2}\right)} \int_{-\infty}^{\infty} \left[ \Phi\left(\frac{\mathrm{i} y}{\sqrt d}\right)^{d+1} + \Phi\left(-\frac{\mathrm{i} y}{\sqrt d}\right)^{d+1} \right] \mathrm{e}^{-y^2/2}\,\mathrm{d}y. $$

In this paper, we prove a monotonicity principle for Beckner inequalities of the form \[ \frac{\mu(f^2)-\mu(f^p)^{2/p}}{2-p}\leq C\E(f,f), \qquad f\in\D(\E),\quad 1\le p\le2, \] where $\E$ is a conservative symmetric Dirichlet form. If the inequality holds for every exponent $p$, it holds with the same constant $C$ for all $q\in[1,p]$. This resolves an open question posed in Chapter~6 of Wang's monograph~\cite{Wangbook}. We also study weak Beckner inequalities of the form \[ \Psi_p(f)\le \beta(r)\E(f,f)+r\,\Osc(f)^2,\qquad f\in\D(\E),\quad r>0, \] where $\beta$ is a rate function. We show that the same monotonicity principle preserves the entire rate function. We further derive consequences for polynomial rates, semigroup contractivity, and concentration.

Emerging 6G industrial IoT architectures require wireless networked control systems capable of stabilizing diverse control loops over tightly constrained radio resources. Conventional periodic and Age-of-Information (AoI) based scheduling guarantees bounded staleness at the cost of persistent channel saturation. Conversely, pure event-triggered (PureET) strategies minimize transmissions but risk catastrophic silent deterioration when local sensor-side thresholds fail to reflect critical state evolution. To bridge this gap, we propose a communication-control co-design framework governed by a 6G Semantic Layer that independently arbitrates uplink and downlink resources. Instead of relying on freshness, our architecture evaluates the actual control impact of a packet using the state-to-error ratio (SER). We unify this control confidence with channel reliability in terms of signal-to-noise ratio (SNR) to orchestrate a threshold-based sensor push and a state-aware controller pull mechanism. To ensure equitable resource allocation across dynamically heterogeneous plants, the proposed framework explicitly scales actuation deadbands according to local plant dynamics. Simulations over Rayleigh-faded channels demonstrate that this approach fundamentally shifts the Pareto frontier between transmission rate and control quality. The proposed scheme achieves tracking accuracy comparable to periodic schedulers at a reduced communication overhead, while mitigating the estimation errors characteristic of PureET.

Filter bubbles can be understood as configurations characterized by within-group proximity and separation from outsiders. Personalized feeds may form or preserve such configurations. We develop hybrid-style logics for reasoning about filter bubbles. The static logic characterizes bubble-shaped configurations, while its dynamic extension captures their formation and persistence under personalized feeds. We provide sound and strongly complete axiomatizations for the static and dynamic logics and prove their decidability.

We prove the following inequality for Boolean functions: $2\sum_{x\in F_2^n}f(x)wt(x)\geq wt(f)(n-deg(f))$. Using this inequality, we establish the Fourier Entropy-Influence (FEI) conjecture for Boolean plateaued functions. In particular, we show that the sharp FEI constant for the class of plateaued functions is 4. We also prove the FEI conjecture for partially bent functions and show that the corresponding sharp constant is 2. Finally, we derive several estimates for the p-biased distribution on the Boolean hypercube. Keywords: Fourier entropy, total influence, average sensitivity, plateaued function, algebraic degree, Reed-Muller code, p-biased distribution.

We show that, for a second-countable locally compact Hausdorff étale minimal groupoid with compact unit space, simplicity of the reduced groupoid C*-algebra implies the existence of a comeager set of unit points with C*-simple isotropy group. Combining this result with work of Christensen and Neshveyev on exotic completions of isotropy group algebras, we show that the converse implication is also true. Finally, we construct a Hausdorff étale minimal groupoid with an isotropy group whose induced exotic completion differs from its reduced group C*-algebra, answering a question of Christensen and Neshveyev.

We study local convergence to the dynamical $\Phi^4_3$ model under a vanishing quintic perturbation. More precisely, on the three-dimensional torus we consider $$\partial_tu_\varepsilon=\Delta u_\varepsilon-\varepsilon^\alpha u_\varepsilon^5+\xi_\varepsilon+C_\varepsilon u_\varepsilon+\widetilde C_\varepsilon u_\varepsilon^3$$ for $\alpha\in(\frac56,1)$, where $\xi_\varepsilon$ is a spatial mollification of space-time white noise. Although the quintic coefficient vanishes, its contractions generate divergent linear and cubic contributions. We identify suitable mass and cubic counterterms that compensate these divergences. Using paracontrolled calculus, we construct the required enhanced stochastic data and prove that their renormalized higher-order components vanish, while the remaining coordinates converge to the enhanced data of the dynamical $\Phi^4_3(\lambda)$ model. We further establish local well-posedness and stability of the associated deterministic solution map. Consequently, for well-prepared initial conditions, $u_\varepsilon$ converges in probability, as a random local solution germ, to the renormalized dynamical $\Phi^4_3(\lambda)$ solution. In particular, the cubic counterterm allows convergence strictly below the threshold $\alpha=1$ arising when only linear renormalization is allowed.