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1. Realizing trees of configurations in thin sets

   https://doi.org/10.2140/pjm.2025.335.355  

   Greenleaf, Allan; Iosevich, Alex; Taylor, Krystal (January 2025, Pacific Journal of Mathematics)

   Let $$\phi(x,y)$$ be a continuous function, smooth away from the diagonal, such that, for some $$\alpha>0$$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^{\phi}f(x)=\int_{\phi(x,y)=t} f(y) \psi(y) d\sigma_{x,t}(y) \end{equation} map $$L^2({\mathbb R}^d) \to H^{\alpha}({\mathbb R}^d)$$ for all $t>0$. Let $$E$$ be a compact subset of $${\mathbb R}^d$$ for some $$d \ge 2$$, and suppose that the Hausdorff dimension of $$E$$ is $$>d-\alpha$$. We show that any tree graph $$T$$ on $k+1$ ($$k \ge 1$$) vertices is realizable in $$E$$, in the sense that there exist distinct $$x^1, x^2, \dots, x^{k+1} \in E$$ and $t>0$ such that the $$\phi$$-distance $$\phi(x^i, x^j)$$ is equal to $$t$$ for all pairs $(i,j)$ corresponding to the edges of the graph $$T$$. 

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2. Diffusions interacting through a random matrix: universality via stochastic Taylor expansion

   https://doi.org/10.1007/s00440-021-01027-7  

   Dembo, Amir; Gheissari, Reza (August 2021, Probability Theory and Related Fields)

   Abstract Consider $$(X_{i}(t))$$ ( X i ( t ) ) solving a system of N stochastic differential equations interacting through a random matrix $${\mathbf {J}} = (J_{ij})$$ J = ( J ij ) with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $$(X_i(t))$$ ( X i ( t ) ) , initialized from some $$\mu $$ μ independent of  $${\mathbf {J}}$$ J , are universal, i.e., only depend on the choice of the distribution $$\mathbf {J}$$ J through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks. 

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3. A Generalization of the Bollobás Set Pairs Inequality

   https://doi.org/10.37236/9627  

   O'Neill, Jason; Verstraete, Jacques (July 2021, The Electronic Journal of Combinatorics)

   The Bollobás set pairs inequality is a fundamental result in extremal set theory with many applications. In this paper, for $$n \geqslant k \geqslant t \geqslant 2$$, we consider a collection of $$k$$ families $$\mathcal{A}_i: 1 \leq i \leqslant k$$ where $$\mathcal{A}_i = \{ A_{i,j} \subset [n] : j \in [n] \}$$ so that $$A_{1, i_1} \cap \cdots \cap A_{k,i_k} \neq \varnothing$$ if and only if there are at least $$t$$ distinct indices $$i_1,i_2,\dots,i_k$$. Via a natural connection to a hypergraph covering problem, we give bounds on the maximum size $$\beta_{k,t}(n)$$ of the families with ground set $[n]$. 

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4. Stable Big Bang formation for Einstein’s equations: The complete sub-critical regime

   https://doi.org/10.1090/jams/1015  

   Fournodavlos, Grigorios; Rodnianski, Igor; Speck, Jared (July 2023, Journal of the American Mathematical Society)

   For $(t,x)\in <\#comment/>(0,\mathrm{\infty <\#comment/>})\times <\#comment/>{\mathbb{T}}^{\mathfrak{D}}$, the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as $t\downarrow <\#comment/>0$, i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents ${\stackrel{~<\#comment/>}{q}}_1,\cdots <\#comment/>,{\stackrel{~<\#comment/>}{q}}_{\mathfrak{D}}\in <\#comment/>\mathbb{R}$, which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be dynamically stable, that is, stable under perturbations of the Kasner initial data, given say at $\{t=1\}$, as long as the exponents are “sub-critical” in the following sense: $\underset{\begin{array}{c}I,J,B=1,\cdots <\#comment/>,\mathfrak{D}\\ I>J\end{array}}{max}\{{\stackrel{~<\#comment/>}{q}}_I+{\stackrel{~<\#comment/>}{q}}_J-<\#comment/>{\stackrel{~<\#comment/>}{q}}_B\}>1$. Previous works have rigorously shown the dynamic stability of the Kasner Big Bang singularity under stronger assumptions: (1) the Einstein-scalar field system with $\mathfrak{D}=3$and ${\stackrel{~<\#comment/>}{q}}_1\approx <\#comment/>{\stackrel{~<\#comment/>}{q}}_2\approx <\#comment/>{\stackrel{~<\#comment/>}{q}}_3\approx <\#comment/>1/3$, which corresponds to the stability of the Friedmann–Lemaître–Robertson–Walker solution’s Big Bang or (2) the Einstein-vacuum equations for $\mathfrak{D}\ge <\#comment/>38$with $\underset{I=1,\cdots <\#comment/>,\mathfrak{D}}{max}|{\stackrel{~<\#comment/>}{q}}_I|>1/6$. In this paper, we prove that the Kasner singularity is dynamically stable forallsub-critical Kasner exponents, thereby justifying the heuristics in the literature in the full regime where stable monotonic-type curvature-blowup is expected. We treat in detail the $1+\mathfrak{D}$-dimensional Einstein-scalar field system for all $\mathfrak{D}\ge <\#comment/>3$and the $1+\mathfrak{D}$-dimensional Einstein-vacuum equations for $\mathfrak{D}\ge <\#comment/>10$; both of these systems feature non-empty sets of sub-critical Kasner solutions. Moreover, for the Einstein-vacuum equations in $1+3$dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have dynamically stable Big Bangs under polarized $U\left(1\right)$-symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized $U\left(1\right)$-symmetric solutions. Our proof relies on a new formulation of Einstein’s equations: we use a constant-mean-curvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi–Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent sub-criticality condition for the dynamic stability of the flow: the condition leads to the time-integrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to $t$, and to handle this difficulty, we use $t$-weighted high order energies, and we control non-linear error terms by exploiting monotonicity induced by the $t$-weights and interpolating between the singularity-strength of the solution’s low order and high order derivatives. Finally, we note that our formulation of Einstein’s equations highlights the quantities that might generate instabilities outside of the sub-critical regime. 

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5. Painlevé-III Monodromy Maps Under the $$D_6\to D_8$$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions

   https://doi.org/10.3842/SIGMA.2024.019  

   Barhoumi, Ahmad; Lisovyy, Oleg; Miller, Peter D; Prokhorov, Andrei (March 2024, Symmetry, Integrability and Geometry: Methods and Applications)

   The third Painlevé equation in its generic form, often referred to as Painlevé-III($$D_6$$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x} \right)^2-\frac{1}{x} \frac{{\rm d}u}{{\rm d}x} + \frac{\alpha u^2 + \beta}{x}+4u^3-\frac{4}{u}, \qquad \alpha,\beta \in \mathbb C. $$ Starting from a generic initial solution $$u_0(x)$$ corresponding to parameters $$\alpha$$, $$\beta$$, denoted as the triple $$(u_0(x),\alpha,\beta)$$, we apply an explicit Bäcklund transformation to generate a family of solutions $$(u_n(x),\alpha + 4n,\beta + 4n)$$ indexed by $$n \in \mathbb N$$. We study the large $$n$$ behavior of the solutions $$(u_n(x), \alpha + 4n, \beta + 4n)$$ under the scaling $x = z/n$ in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution $$u_n(z/n)$$. Our main result is a proof that the limit of solutions $$u_n(z/n)$$ exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III($$D_8$$), $$ \frac{{\rm d}^2U}{{\rm d}z^2} =\frac{1}{U}\left(\frac{{\rm d}U}{{\rm d}z}\right)^2-\frac{1}{z} \frac{{\rm d}U}{{\rm d}z} + \frac{4U^2 + 4}{z}.$$ A notable application of our result is to rational solutions of Painlevé-III($$D_6$$), which are constructed using the seed solution $(1,4m,-4m)$ where $$m \in \mathbb C \setminus \big(\mathbb Z + \frac{1}{2}\big)$$ and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at $z = 0$ when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both $$D_6$$ and $$D_8$$ at $z = 0$. We also deduce the large $$n$$ behavior of the Umemura polynomials in a neighborhood of $z = 0$. 

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