Abstract The paper studies the existence of solutions for certain types of nonlinear heat transfer equations and nonlinear Navier–Stokes equations based on the theory of p-regularity and its most important result, namely the Generalized Lyusternik Theorem. We present this theorem using a sequence of supporting lemmas and provide a detailed procedure for finding solutions to the above equations, along with explicit formulas for these solutions.
We investigate the average behaviour of the Fourier coefficients of symmetric square $L$-functions, $\lambda _{\operatorname{sym}<^>2 f}(n)$, attached to holomorphic Hecke eigenforms. Recent work by Tiwari and Godara (P. Tiwari and N. K. Godara, On the power moments of hybrid arithmetic functions associated with the Hecke eigenvalues, Ramanujan J. 66 (2025), 46) has studied related sums over sequences generated by polynomials $g(\underline{x})$ and $h(\underline{x})$. We extend this line of inquiry by analysing the $m$-th power sums over these same two polynomial sequences, $\sum _{n=g(\underline{a})+1 \le x} \lambda _{\operatorname{sym}<^>2 f}(n<^>m)$ and $\sum _{n=h(\underline{a})+1 \le x} \lambda _{\operatorname{sym}<^>2 f}(n<^>m)$, alongside the sum $\sum _{n=\mathcal {Q}(\underline{x}) \le x} \lambda _{\operatorname{sym}<^>2 f}(n<^>m)$ over integers $n$ represented by primitive positive-definite binary quadratic forms. In all three cases, the underlying multiplicative representation function $r(n)$ is related to coefficients of Eisenstein series. We establish unified asymptotic formulas for all $m \ge 2$ for all three of these sequences.
In this paper, we establish a new quantitative result for the simultaneous non-vanishing of the central values of quadratically twisted L-functions associated with $SL(2, \mathbb {Z})$ and $SL(3, \mathbb {Z})$ Hecke-Maass cusp forms. Our main result is derived from the calculation of moments of twisted central L-values, which is accomplished through the application of a Waldspurger-type formula for Maass cusp forms and the Kuznetsov trace formula for the Kohnen subspace.
Let $X \ge y \ge 2$, and let $u = \frac{\log X}{\log y}$. We say a number is y-smooth if all of its prime factors are less than or equal to $y$. In this paper, we study the distribution of y-smooth numbers in short intervals. In particular, for $y \ge \exp \left( (\log X)<^>{2/3 + \varepsilon } ight)$, we show that the interval $[x, x+h]$ contains a y-smooth number for almost all $x \in [X, 2X]$, provided $h \ge \exp \left( (1 + \varepsilon ) \left( \frac{11}{8} u \log u + 4 \log \log X ight) ight)$, and $X$ is sufficiently large depending on $\varepsilon$. This result improves upon an earlier result by Matom & auml;ki. Additionally, we provide the corresponding 'all intervals' type result. Our approach relies on a strategically factorized Dirichlet polynomial, much like the earlier work of Matom & auml;ki. The improvement in our results stems from the integration of ideas introduced in the breakthrough work of Matom & auml;ki and Radziwi & lstrok;& lstrok;.
We prove one-level density results for L-functions attached to primitive forms of level q, averaged over square-free q, conditional on the Generalized Riemann Hypothesis (GRH). We treat the even and odd orthogonal families separately and extend the support of the Fourier transform of the test function to (-3,3). This extended support yields the strongest known non-vanishing results for these families of L-functions and their derivatives at the central point, conditional on GRH.
This paper explores various differentiable structures on the product manifold $M \times \mathbb {S}<^>k$, where M is either a 4-dimensional closed, oriented, smooth manifold or a simply connected 5-dimensional closed, smooth manifold. We identify the possible stable homotopy types of M and use it to calculate the concordance inertia group and the concordance structure set of $M\times \mathbb {S}<^>k$ for $1\le k\le 10.$ These calculations enable us to further classify all manifolds that are homeomorphic to $\mathbb {C}P<^>2\times \mathbb {S}<^>k$, up to diffeomorphism, for each $4\le k\le 6$.
The Littlewood conjecture, proven by Konyagin and McGehee-Pigno-Smith in the 1980s, states that if $A\subset \mathbb {Z}$ is a finite set of integers with $\vert A\vert =N$, then $\Vert \widehat{1_A}\Vert _1\geqslant c\log N$ for some absolute constant $c \gt 0$. We explore what structure A must have if $\Vert \widehat{1_A}\Vert _1\leqslant K\log N$ for some constant K. Under such an assumption, we prove, for instance, that A contains a subset $A<^>{\prime }\subseteq A$ with $\vert A<^>{\prime }\vert \geqslant N<^>{0.99}$ such that $|A<^>{\prime } + A<^>{\prime }| \ll K<^>{O(1)}|A<^>{\prime }|$. As a consequence, for any $k\geqslant 3$, if N is sufficiently large depending on k and K, then A must contain an arithmetic progression of length k. A byproduct of our analysis is a (slightly) improved bound for the constant c.
ABSTRACT Every composition of two strictly singular operators is compact on the Baernstein space $B_p$ for $1 \lt p \lt \infty$ and on the p-convexified Schreier space $S_{p}$ for $1 \leqslant p \lt \infty$. Furthermore, every subsymmetric basic sequence in $B_p$ (respectively, $S_p$) is equivalent to the unit vector basis for $\ell _p$ (respectively, $c_0$), and the Banach spaces $B_p$ and $S_p$ contain block basic sequences whose closed span is not complemented.
We show that any locally planar tropical curve Γ⊂ℝ^n (with unit edge weights) can be realized as the limit of the rescaled moment map images of a family of special Lagrangian submanifolds in T^*T^n with respect to the Euclidean structure. This is based on a gluing construction that matches special Lagrangian local models to the combinatorics of Γ, thereby establishing a direct link between tropical geometry and special Lagrangian geometry.
For any positive integer k, let $X_k$ be a projective irreducible nodal curve with k nodes. We show that the Betti numbers and the mixed Hodge numbers of the compactified Jacobian $\overline{J_k}$ of an irreducible nodal curve $X_k$ with k nodes are the same as the Betti numbers and the mixed Hodge numbers of $J_0 imes R<^>k$, where $J_0$ is the Jacobian of the normalization of the irreducible nodal curve and R denotes the rational nodal curve with one node. We prove it by constructing a trivial topologically locally family of projective varieties that contain both $\overline{J_k}$ and $J_0 imes R<^>k$ as fibers. Explicit formulas for the Betti and Hogde numbers are obtained by applying the Kunneth type formula on the Betti and Hodge numbers of the product space $J_0 imes R<^>k$.
We define categories of stratified manifolds (s-manifolds) and stratified manifolds with corners (s-manifolds with corners). An s-manifold ${\boldsymbol{X}}$ of dimension n is a Hausdorff, locally compact topological space X with a stratification $X=\coprod _{i\in I}X<^>i$ into locally closed subsets $X<^>i$ which are smooth manifolds of dimension $\leqslant n$, satisfying some conditions. S-manifolds can be very singular, but still share many good properties with ordinary manifolds, for example, an oriented s-manifold ${\boldsymbol{X}}$ has a fundamental class $[{\boldsymbol{X}}]_{\rm fund}$ in Steenrod homology $H_n<^>{\rm St}(X,{\mathbin {\mathbb {Z}}})$, and transverse fibre products exist in the category of s-manifolds. S-manifolds are designed for applications in Symplectic Geometry. In future work we hope to show that after suitable perturbations, the moduli spaces ${\mathbin {\smash{\, \, \overline{\!\!\mathcal {M}\!}\, }}\vphantom{\cal M}}$ of J-holomorphic curves used to define Gromov-Witten invariants, Lagrangian Floer cohomology, Fukaya categories, and so on, can be made into s-manifolds or s-manifolds with corners, and their fundamental classes used to define Gromov-Witten invariants, Lagrangian Floer cohomology.
We investigate the mean value of the inner product of squared $\mathrm{GL}_{n}$ degenerate maximal parabolic Eisenstein series against a smooth compactly supported function lying in a restricted space of incomplete Eisenstein series induced from an $\mathrm{SL}_{2}(\mathbb {Z})$ Hecke-Maa ss cusp form $\varphi$. Our result breaks the fundamental threshold with a polynomial power-saving beyond the pointwise implications of the generalized Lindel & ouml;f hypothesis for L-functions attached to $\varphi$. Furthermore, we evaluate the Archimedean quantum variance and establish approximate orthogonality, expanding upon Zhang's work on quantum unique ergodicity for $\mathrm{GL}_{n}$ degenerate maximal parabolic Eisenstein series as well as Huang's work on quantum variance for $\mathrm{GL}_{2}$ Eisenstein series. Despite the theoretical strength of these manifestations, our argument relies exclusively on the Watson-Ichino-type formula for incomplete Eisenstein series of type $(2, 1, \ldots , 1)$ and Jutila's asymptotic formula for the second moment of L-functions attached to $\varphi$ in long intervals, supplemented by a standard analytical toolbox.
Let F be a number field with the adele ring ${\mathbb {A}}_F$; $\pi _1, \pi _2$ be two fixed unitary cuspidal automorphic representations of $\mathrm{PGL}_2({\mathbb {A}}_F)$ with finite coprime conductors ${\mathfrak {u}}$ and ${\mathfrak {v}}$, respectively; and ${\mathfrak {q}},{\mathfrak {l}}$ be two coprime integral ideals with $({\mathfrak {q}}{\mathfrak {l}}, {\mathfrak {u}}{\mathfrak {v}})=1$. Following the work of R. Zacharias, we estimate the first moment of $L(\frac{1}{2}, \pi \otimes \pi _1 \otimes \pi _2)$ twisted by the Hecke eigenvalues $\lambda _{\pi }({\mathfrak {l}})$, where $\pi$ runs through unitary automorphic representations with finite conductors dividing ${\mathfrak {u}}{\mathfrak {v}}{\mathfrak {q}}$. By applying the triple product integrals, spectral decomposition and the Plancherel formula, we get a reciprocity formula that links the twisted first moment of triple product L-functions to the spectral expansion of certain triple product periods over automorphic representations with finite conductors dividing ${\mathfrak {l}}$. As an application, we study the subconvexity problem for the triple product L-functions in the level aspect and give a subconvexity bound for $L(\frac{1}{2}, \pi \otimes \pi _1 \otimes \pi _2)$ in terms of the norm of ${\mathfrak {q}}$.
Let $p\geq 5$ be a prime number. Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$. Let $K$ be an imaginary quadratic field where $p$ splits, and such that the generalized Heegner hypothesis holds. Under mild hypotheses, we show that if the $p$-adic height of the Heegner point of $\mathsf{E}$ over $K$ is non-zero, then Mazur's conjecture on the growth of Selmer coranks in the $\mathbb{Z}_p^2$-extension of $K$ holds.
We establish nontrivial bounds for bilinear sums involving the Möbius function evaluated over solutions to a broad class of equations. Several of our results may be regarded as Möbius-function analogues of the ternary Goldbach problem. By contrast, the binary versions of our results remain out of reach, much like the binary Goldbach problem. Nevertheless, we make partial progress in this direction by restricting the range of the third variable as far as possible.
We study the negative band number of braids, knots, and links using Birman, Ko, and Lee's left-canonical form of a braid. As applications, we characterize up to conjugacy strongly quasipositive braids and almost strongly quasipositive braids.
We describe the cohomological Hall algebra of torsion sheaves on a weighted projective line with weights $(2, \dots, 2)$ in terms of generators and relations.
Let f be a holomorphic Hecke cusp form of weight k for SL2(Z ), and let (lambdaf(n))n >= 1 denote its sequence of Hecke eigenvalues. We compute the first and second moments of the sums S(x, f) = Sigmax lambdaf(n), of large weight , in the regime where the length of the sums x is smaller than k2. We observe transitions in the size of the sums when x approximate to k and x approximate to k2. In subsequent work (part II), it will be shown that once is larger than k2(where the latter transition occurs), the average size of the sums S(x,f) becomes dramatically smaller.
Under natural assumptions we find local normal forms for generalized complex structures on transitive Courant algebroids, which extend Gualtieri's Darboux theorem for generalized complex structures on manifolds. When the base of the Courant algebroid is a point, they reduce to Wang's description of invariant complex structures on compact semisimple Lie groups.
