Abstract This article investigates coefficient randomizations of holomorphic functions and the resulting random analytic functions within the framework of mixed norm spaces. Adopting the perspective of the random symbol space ( X ) ⋆ = { f ∈ H ( B n ) : P ( R f ∈ X ) = 1 } $(\mathcal {X})_{\star }=\{f\in H(\mathbb {B}_n): \mathbb {P}(\mathcal {R} f\in \mathcal {X})=1\}$ left parenthesis script upper X right parenthesis Subscript star Baseline equals StartSet f element of upper H left parenthesis double struck upper B Subscript n Baseline right parenthesis colon double struck upper P left parenthesis script upper R f element of script upper X right parenthesis equals 1 EndSet , we first focus on the unit disk D $\mathbb {D}$ double struck upper D . Specifically, for q ≥ 4 $q\geq 4$ q greater than or equals 4 , we identify the conditions under which the random symbol space of the mixed-norm space H p , q , u ( D ) $H^{p,q,u}(\mathbb {D})$ upper H Superscript p comma q comma u Baseline left parenthesis double struck upper D right parenthesis coincides with H 2 , q , u ( D ) $H^{2,q,u}(\mathbb {D})$ upper H Superscript 2 comma q comma u Baseline left parenthesis double struck upper D right parenthesis . Furthermore, we extend our analysis to the higher-dimensional unit ball B n $\mathbb {B}_n$ double struck upper B Subscript n . We establish sufficient conditions and necessary constraints linking the deterministic membership of a function to the almost sure membership of its randomization in mixed norm spaces. A key finding is the identification of a sharp, dimension-dependent weight loss phenomenon that arises for n > 1 $n>1$ n greater than 1 .
An S k $S_k$ upper S Subscript k -set in a group Gamma $\Gamma $ normal upper Gamma is a set A subset of Gamma $A\subseteq \Gamma $ upper A subset of or equal to normal upper Gamma such that alpha 1 & mldr; alpha k = beta 1 & mldr; beta k $\alpha _1\dots \alpha _k=\beta _1\dots \beta _k$ alpha 1 ellipsis alpha Subscript k Baseline equals beta 1 ellipsis beta Subscript k Baseline with alpha i , beta i is an element of A $\alpha _i,\beta _i\in A$ alpha Subscript i Baseline comma beta Subscript i Baseline element of upper A implies ( alpha 1 , & mldr; , alpha k ) = ( beta 1 , & mldr; , beta k ) $(\alpha _1,\ldots ,\alpha _k)=(\beta _1,\ldots ,\beta _k)$ left parenthesis alpha 1 comma ellipsis comma alpha Subscript k Baseline right parenthesis equals left parenthesis beta 1 comma ellipsis comma beta Subscript k Baseline right parenthesis . An S k ' $S_k'$ upper S prime Subscript k -set is a set such that alpha 1 beta 1 - 1 & mldr; alpha k beta k - 1 = 1 $\alpha _1\beta _1<^>{-1}\dots \alpha _k\beta _k<^>{-1}=1$ alpha 1 beta 1 Superscript negative 1 Baseline ellipsis alpha Subscript k Baseline beta Subscript k Superscript negative 1 Baseline equals 1 implies that there exists i such that alpha i = beta i or beta i = alpha i + 1 $\alpha _i=\beta _i ext { or }\beta _i=\alpha _{i+1}$ alpha Subscript i Baseline equals beta Subscript i Baseline or beta Subscript i Baseline equals alpha Subscript i plus 1 Baseline . We give explicit constructions of large S k $S_k$ upper S Subscript k -sets in the groups Sym ( n ) $\mathrm {Sym}(n)$ upper S y m left parenthesis n right parenthesis and Alt ( n ) $\mathrm {Alt}(n)$ upper A l t left parenthesis n right parenthesis and S 2 $S_2$ upper S 2 -sets in Sym ( n ) & times; Sym ( n ) $\mathrm {Sym}(n) imes \mathrm {Sym}(n)$ upper S y m left parenthesis n right parenthesis times upper S y m left parenthesis n right parenthesis and Alt ( n ) & times; Alt ( n ) $\mathrm {Alt}(n) imes \mathrm {Alt}(n)$ upper A l t left parenthesis n right parenthesis times upper A l t left parenthesis n right parenthesis . We give probabilistic constructions which yield large S 2 ' $S_2'$ upper S prime 2 -sets in Sym ( n ) $\mathrm {Sym}(n)$ upper S y m left parenthesis n right parenthesis . We also give upper bounds on the size of S k $S_k$ upper S Subscript k -sets in certain groups, improving the trivial bound by a constant multiplicative factor. We describe some connections between S k $S_k$ upper S Subscript k -sets and extremal graph theory. In particular, we determine up to a constant factor the minimum outdegree of a digraph which guarantees even cycles with certain orientations. As applications, we improve the upper bound on Hamilton paths which pairwise create a two-part cycle of given length, and we show that a directed version of the Erd & odblac;s-Simonovits compactness conjecture is false.
This article focuses on the occurrence of 3-point configurations in subsets of $\mathbb {R}^d$ of sufficient thickness. We prove that a compact set $A\subset \mathbb {R}^d$ contains a similar copy of any linear $3$ -point configuration (such as a $3$ -point arithmetic progression) provided that A satisfies a mild Yavicoli-thickness condition and an r -uniformity condition for $d\geq 2$ ; or, when $d=1$ , the result holds provided that the Newhouse thickness of A is at least $1$ . Moreover, we prove that compact sets $A\subset \mathbb {R}^2$ contain the vertices of an equilateral triangle (and more generally, the vertices of a similar copy of any given triangle) provided A satisfies a mild Yavicoli-thickness condition and an r -uniformity condition. Further, $C\times C$ contains the vertices of an equilateral triangle (and more generally the vertices of a similar copy of any given 3-point configuration) provided the Newhouse thickness of C is at least $1$ . These are among the first results in the literature to give explicit criteria for the occurrence of 3-point configurations in the plane.
In their 2016 paper on exotic Bailey–Slater SPT-functions, Garvan and Jennings-Shaffer introduced many new spt-crank-type functions and proposed a conjecture that the spt-crank-type functions $M_{C1}(m,n)$ and $M_{C5}(m,n)$ are both nonnegative for all $m\in \mathbb {Z}$ and $n\in \mathbb {N}.$ Applying Wright’s circle method, Jang and Kim showed that $M_{C1}(m,n)$ and $M_{C5}(m,n)$ are both positive for a fixed integer m and large enough integers $n.$ Up to now, no complete proof of this conjecture has been given. In this article, we provide a complete proof for this conjecture by using the theory of lattice points. Our proof is quite different from that of Jang and Kim.
The main result of this article is the construction of a new class of weight shifting operators, similar to the theta operators of de Shalit and Goren (2019, Algebra NumberTheory, 13, 1829-1877), Eischen et al. (2021, Algebra Number Theory, 15, 1469-1504), and others, which are defined on the lower Ekedahl-Oort strata of the geometric special fiber of unitary Shimura varieties of signature (n - 1, 1) at a good prime p, split in the reflex field E, which we assume to be quadratic imaginary. These operators act on certain graded sheaves which are obtained from the arithmetic structure of the EO strata, in particular the p-rank on each stratum. We expect these operators to have applications to the study of Hecke-eigensystems of (mod p) modular forms and generalizations of the weight part of Serre's conjecture.
We give a model-independent definition of limits for diagrams valued in an $(\infty ,n)$ -category. We show that this definition is compatible with the existing notion of homotopy $2$ -limits for $2$ -categories, with the existing notion of $(\infty ,1)$ -limits for $(\infty ,1)$ -categories, and with itself across different values of n.
As noted by Landau and Lifshitz in their work Course of Theoretical Physics (Vol. 6, Pergamon Press, Oxford, 2nd edition, 1987), "if the relaxation time of these processes is long, a considerable dissipation of energy occurs when the fluid is compressed or expanded," then the volume viscosity coefficient becomes relatively large, making the shear viscosity coefficient much smaller in comparison. In this article, we take this phenomenon into consideration and investigate isentropic compressible magnetohydrodynamic flow with zero shear viscosity in a periodic domain. We prove the global existence of smooth solutions when the initial data are close to a background magnetic field. In addition, stability and large-time decay rates are also obtained. Mathematically, the zero shear viscosity makes the elliptic operator " $\mu \Delta + u abla ext{div}\, $ " lose its uniform ellipticity, rendering the classical dissipation mechanisms for velocity inapplicable. The stabilizing effect of the background magnetic field plays a crucial role in our analysis.
In Benedetto et al. (2025, J. Lond. Math. Soc., 112, Article no. e70257), we provided an explicit description of the arboreal Galois group of the postcritically finite polynomial $f(z) = z<^>2 +c$ in the special case when the critical point $0$ is periodic under the action of $f(z)$ . In the current article, we complete the picture for all postcritically finite polynomials by addressing the cases when $0$ is strictly preperiodic for the polynomial $f(z)$ .
In this article, we investigate a high-order quasilinear hyperbolic equation that involves Kirchhoff damping and logarithmic source: u(tt)-div(divided by del u divided by(p-2)del u) +sigma(parallel to del u parallel to(2)(2))u(t) +Delta(2)u = divided by u divided by(q-2)u log divided by u divided by, in Omega & times; (0, T-max), subject to null Neumann boundary value conditions, where Omega subset of R-n is an open bounded domain with smooth boundary and p, q >2 and sigma(parallel to del u parallel to(2)(2)) is the Kirchhoff-type coefficient of the damping u(t). Through the utilization of the Faedo-Galerkin approximation, we gain the well-posedness of local weak solutions. When q >= p, we construct algebraic and exponential decay estimates for the energy of global weak solutions. At the same time, relying on the contradiction argument and the auxiliary function method, we prove that the weak solutions blow-up with negative initial energy for q > p. When q < p, we achieve that the weak solutions are globally bounded.
Graph burning is a discrete process that models the spread of influence through a network using a fire as a proxy for the type of influence being spread. This process was recently extended to apply to hypergraphs in both round-based and lazy settings. We introduce a variant of hypergraph burning that uses an alternative propagation rule for how the fire spreads - if some fixed proportion of vertices are on fire in a hyperedge, then in the next round, the entire hyperedge catches fire.We obtain bounds on the burning numbers of general hypergraphs, and introduce the concept of the burning distribution, which describes how the burning numbers change as the proportion parameter ranges over $(0,1)$ . We also obtain computational results which suggest there is a strong correlation between the automorphism group order and the lazy burning number of a balanced incomplete block design.
We investigated the symplectic geometry of homogeneous spaces associated with semisimple Lie groups, focusing on cotangent bundles of maximal flag manifolds. Our work provides an explicit description of the canonical symplectic structure on these spaces using connections and curvatures of principal bundles naturally associated with the underlying Lie groups. We extend classical results concerning the exactness of symplectic forms on adjoint orbits, previously known for specific Lie algebras, to arbitrary simple Lie groups. In particular, we identify conditions under which the Kostant-Kirillov-Souriau form on a regular adjoint orbit coincides with the canonical symplectic form of the cotangent bundle, yielding exact symplectic structures. The approach combines differential-geometric techniques with Lie-theoretic constructions, offering a unifying framework that connects the geometry of coadjoint orbits with symplectic structures on homogeneous spaces.
This article studies the stability of the L(p )torsional measure in dimension n >= 3. We show that the ball is the unique domain for the solution to the related overdetermined boundary value problem. Two concepts of relative asymmetry distance and Lc-distance have been used to estimate the stability of the L(p )torsional measure, based on different stability results of the L-p -width functionals. The result has been uniformly given via relative asymmetry distance for all 1 < p # n + 2. In terms of Lc-distance, the result has been split into two cases: p is an element of (1, n), and n < p # n + 2.
The theta cycle of a modular form modulo a prime p >= 5 is well understood. By contrast, the theta cycle modulo a power of p is still mysterious and experimentally erratic. Here, we completely determine the theta cycle of a weight k[ (p-k+1)/2 & RightFloor; further low points at regular positions. Moreover, we detect low points at exceptional positions which solve a quadratic equation modulo p, and which disturb the otherwise regular structure in the segments that we exhibit.
We investigate the qualitative properties of positive solutions to mixed local-nonlocal equations with indefinite nonlinearities, emphasizing the interaction between classical and fractional Laplacians. We first establish maximum principles and prove strict monotonicity along the $x_1$ -direction for mixed elliptic operators. By combining a mollified first eigenfunction with a suitable sub-solution, we derive nonexistence results for the mixed operator ${(-\Delta )}<^>s - \Delta $ via a contradiction argument. These results are further extended to the parabolic setting, incorporating both the Marchaud-type fractional time derivative and the classical first-order derivative, revealing new qualitative features under dual nonlocality. A key aspect of our approach is a careful adaptation of the method of moving planes to the mixed local-nonlocal context. By addressing the distinct scaling behaviors of local and nonlocal terms, the method yields monotonicity and Liouville-type results without standard decay assumptions and provides a framework potentially applicable to a broader class of mixed elliptic and parabolic problems.
We prove that the Weil representation over a non-archimedean local field can be realized with coefficients in a number field. We give an explicit descent argument to describe precisely which number field the Weil representation descends to. Our methods also apply over more general coefficient fields, such as $\ell $ -modular coefficient fields, as well as coefficient rings, such as rings of integers, that is, in families. We also prove that the theta correspondence over a perfect field is valid if and only if it is valid over the algebraic closure of this perfect field. These two results together show that the classical local theta correspondence is rational in the sense that it can be defined over a number field and it is compatible to Galois automorphism.
We study the subsymmetric basic sequence structure of variable exponent Lebesgue spaces L P $L_{\boldsymbol {P}}$ upper L Subscript bold italic upper P built from index functions P : Omega -> ( 0 , infinity ] $\boldsymbol {P}\colon \Omega o (0,\infty ]$ bold italic upper P colon normal upper Omega right arrow left parenthesis 0 comma infinity right bracket on sigma $\sigma $ sigma -finite measure spaces ( Omega , Sigma , mu ) $(\Omega ,\Sigma ,\mu )$ left parenthesis normal upper Omega comma normal upper Sigma comma mu right parenthesis . Specifically, we prove that if P $\boldsymbol {P}$ bold italic upper P is bounded away from infinity, then any complemented subsymmetric basic sequence of L P $L_{\boldsymbol {P}}$ upper L Subscript bold italic upper P is equivalent to the canonical basis of & ell; r $\ell _r$ script l Subscript r for some r >= 1 $r\ge 1$ r greater than or equals 1 in the essential range of P $\boldsymbol {P}$ bold italic upper P .
The well-known proof of Beurling's Theorem in the Hardy space H-2, which describes all shift-invariant subspaces, rests on calculating the orthogonal projection of the unit constant function onto the subspace in question. Extensions to other Hardy spaces H-P for 0 < p < infinity are usually obtained by reduction to the H & sup2; case via inner-outer factorization of H-P functions. In this article, we instead explicitly calculate the metric projection of the unit constant function onto a shift-Invariant subspace of the Hardy space HP when 1 < p < infinity This problem is equivalent to finding the best approximation in HP of the conjugate of an inner function. In H & sup2;, this approximation is always a constant, but in H-P, when p not equal 2, this approximation turns out to be zero or a non-constant outer function. Further, we determine the exact distance between the unit constant and any shift-invariant subspace and propose some open problems. Our results use the notion of Birkhoff-James orthogonality and Pythagorean inequalities, along with an associated dual extremal problem, which leads to some interesting inequalities. Further consequences shed light on the lattice of shift-Invariant subspaces of H-P, as well as the behavior of the zeros of optimal polynomial approximants in H-P.
