Abstract Investigating a recent positive solution of a conjecture of Grunewald and O’Halloran for complex finite-dimensional nilpotent Lie algebras, we find results of existence and uniqueness for the construction of complex nilpotent Lie algebras of arbitrary dimension via pseudobosonic operators. We involve the theory of the deformation of Lie algebras of Gerstenhaber, in order to prove our main results. There is not a generalized version of the Grunewald–O’Halloran Conjecture when we consider pseudoquonic operators, which specialize to pseudobosonic operators, so we prove results of uniqueness for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> {C^{*}} -algebras of pseudoquonic operators leveraging on different methods of functional analysis and operator theory.

Abstract In this paper, we develop a long exact sequence for the path homology of digraphs, providing a useful tool for computing the path homology of digraphs. One application of this result is the proof of a conjecture proposed by S. Chowdhury, which was initially observed through extensive computational experiments. Another interesting application demonstrates that the path homology of n -dimensional grid-like digraphs is concentrated in dimension ≤ n - 1 {\leq n-1} .

Abstract Let G be an affine or hyperbolic rank 2 Kac–Moody group over a finite field 𝔽 q {\mathbb{F}_{q}} . Let X = X q + 1 {X=X_{q+1}} be the Tits building of G , the ( q + 1 ) {(q+1)} -homogeneous tree, and let Γ be a non-uniform lattice in G . When Γ is a standard parabolic subgroup for the negative BN -pair, we define Eisenstein series on Γ \ X {\Gamma\backslash X} and prove its convergence in a half space using Iwasawa decomposition of the Haar measure on G . A crucial tool is a description of the vertices of X in terms of Iwasawa cells. We also prove meromorphic continuation of the Eisenstein series. This requires us to construct an integral operator on the Tits building X and a truncation operator for the Eisenstein series. We also develop the functional analytic framework necessary for proving meromorphic continuation in our setting, by refining and extending Bernstein’s Continuation Principle.

Abstract Let P : ℝ n × ℝ n → ℝ {P:\mathbb{R}^{n}\times\mathbb{R}^{n}\mathbb{\rightarrow R}} be a polynomial mapping where ℝ n {\mathbb{R}^{n}} is the n -dimensional euclidean space (with n ≥ 2 {n\geq 2} ). Let h : ( 0 , ∞ ) → ℝ {h:(0,\infty)\rightarrow\mathbb{R}} be a measurable function, and let Ω ∈ L 1 ⁢ ( 𝕊 n - 1 ) {\Omega\in L^{1}(\mathbb{S}^{n-1})} be a homogeneous function of degree zero on ℝ n {\mathbb{R}^{n}} with mean value zero over the unit sphere 𝕊 n - 1 {\mathbb{S}^{n-1}} . The aim of this paper is to investigate the long-standing problem concerning the L p {L^{p}} boundedness of the oscillatory singular integral operator T P , Ω , h ⁢ f ⁢ ( x ) = p.v. ⁢ ∫ ℝ n e i ⁢ P ⁢ ( x , y ) ⁢ h ⁢ ( | x - y | ) ⁢ | x - y | - n ⁢ Ω ⁢ ( x - y ) ⁢ f ⁢ ( y ) ⁢ 𝑑 y T_{P,\Omega,h}f(x)=\mbox{p.v.}\int_{\mathbb{R}^{n}}e^{iP(x,y)}h(|x-y|)|x-y|^{-% n}\Omega(x-y)f(y)\,dy under the assumption that Ω ∈ L ⁢ log ⁡ L ⁢ ( 𝕊 n - 1 ) {\Omega\in L\log L(\mathbb{S}^{n-1})} and the radial function h is rough in the sense that it satisfies an integrability condition in the form sup R &gt; 0 ⁡ 1 R ⁢ ∫ 0 R | h ⁢ ( t ) | γ ⁢ 𝑑 t &lt; ∞ \sup_{R&gt;0}\frac{1}{R}\int_{0}^{R}|h(t)|^{\gamma}\,dt&lt;\infty for some γ &gt; 1 . {\gamma&gt;1.}

Abstract The spherical capillary water waves equation describes the motion of an almost spherical water droplet under zero gravity governed by water-air interface tension. Using para-differential calculus on compact Lie groups and homogeneous spaces developed by the author, the system is symmetrized into a quasi-linear dispersive para-differential equation of order 1.5 defined on the 2-sphere. An immediate consequence of this symmetrization is a new proof of local well-posedness of the system under much weaker regularity assumption compared to previous results.

Abstract We study congruences for Eisenstein series on SL 2 ⁢ ( ℤ ) {{\rm SL}_{2}(\mathbb{Z})} modulo p 2 {p^{2}} , where p ≥ 5 {p\geq 5} is prime. It is classically known that all Eisenstein series of weight at least 4 are determined modulo p 2 {p^{2}} by those of weight at most p 2 - p + 2 {p^{2}-p+2} . We prove that up to powers of E p - 1 {E_{p-1}} , each such Eisenstein series is in fact determined modulo p 2 {p^{2}} by a modular form of weight at most 2 ⁢ p - 4 {2p-4} . We also determine E 2 {E_{2}} modulo p 2 {p^{2}} in terms of a modular form of weight p + 1 {p+1} .

Abstract We show that a pair of newforms ( f , g ) (f,g) can be uniquely determined by the product of the central 𝐿-values of their twists. To achieve our goal, we prove an asymptotic formula for the average of the product of the central values of two twisted 𝐿-functions, L ⁢ ( 1 / 2 , f × χ ) ⁢ L ⁢ ( 1 / 2 , g × χ ⁢ ψ ) L(1/2,f\times\chi)L(1/2,g\times\chi\psi) , where ( f , g ) (f,g) is a pair of newforms. The average is taken over the primitive Dirichlet characters 𝜒 and 𝜓 of distinct prime moduli.

Abstract In this note, we give an upper bound on the Hausdorff dimension of removable sets for elliptic and canceling homogeneous differential operators with constant coefficients in the class of bounded functions, using a simple extension of Frostman’s lemma in Euclidean space with an additional power decay.

Abstract We consider the set of m × n {m\times n} matrices with rational entries having numerator and denominator of size at most H and obtain various upper bounds on the number of such matrices of a given rank, or with a given determinant, or a given characteristic polynomial. We also consider similar questions for matrices whose entries are Egyptian fractions.