Let [Formula: see text] be a homogeneous ideal in the polynomial ring [Formula: see text], where [Formula: see text] is an algebraically closed field of characteristic zero. Macaulay’s Theorem provides constraints on the Hilbert function of [Formula: see text] or [Formula: see text] from one degree to the next. Nowadays, the standard quotation of Macaulay’s theorem is [Formula: see text], which is regarding the quotient [Formula: see text] and the combinatorial computation in the formula involves the number [Formula: see text] explicitly. However, the origin statement of Macaulay is in fact regarding the Hilbert function of [Formula: see text] itself and the relevant combinatorics explicitly involves the number of variables (i.e. [Formula: see text]) and does not depend on [Formula: see text]. In this paper, we provide an elementary proof of the equivalence between these two versions of Macaulay’s theorem. The original degree-independent version is more suitable for problems such as those involving sums of polynomial squared norms. Motivated by the Hermitian analogue of Hilbert’s 17th problem, SOS conjecture and proper holomorphic mappings between complex unit balls, some questions lead to the study of Hermitian polynomials [Formula: see text] satisfying [Formula: see text] for some [Formula: see text] and a holomorphic mapping [Formula: see text]. Using Macaulay’s Theorem, we derive new inequalities relating [Formula: see text], [Formula: see text], the signature [Formula: see text] of the coefficient matrix of [Formula: see text], and [Formula: see text] (the rank of [Formula: see text] ) and extend these results to norms of arbitrary signatures, which hold uniformly for all bidegrees of [Formula: see text].
Let Gamma congruent to(& Zopf;/2 & Zopf;)(2) be a Klein four-subgroup in the automorphism group of the exceptional Lie group E-6(2), which does not contain any Cartan involution. We classify all the Klein four-symmetric pairs (E-6(2),E-6(2)(Gamma)), for which we also study the branching laws and verify a conjecture.
We consider the isometries of the complex hyperbolic bidisc, that is, the product space [Formula: see text], where each factor [Formula: see text] denotes the complex hyperbolic plane. We investigate the Dirichlet domain formed by the action of a cyclic subgroup generated by [Formula: see text], where each [Formula: see text] is loxodromic. We prove that such a Dirichlet domain has two sides.
Delta-shock wave interactions are analyzed in the one-dimensional shallow-water system with variable bottom topography. The problem is formulated in a distributional framework that accommodates Dirac delta initial data in all state variables. Explicit solutions are constructed and shown to exhibit coherent, soliton-like behavior across all variables, identifying a subclass of delta-shock waves with dynamics closely analogous to classical solitons.
In this paper, we prove several rigidity results of the Clifford torus as a Lagrangian self-shrinker in [Formula: see text] under some restrictions of the Kähler angle with respect to the twisted complex structure [Formula: see text]. Besides, we discuss the rotational symmetry case.
Let [Formula: see text] be a smooth projective variety with [Formula: see text], and let [Formula: see text] be a smooth projective curve of genus [Formula: see text]. We study the Virasoro constraints for the fiber-degree-zero sector of the Gromov–Witten theory of [Formula: see text]. We prove that, if the full-genus Virasoro constraints hold for [Formula: see text], then the full-genus fiber-degree-zero Virasoro constraints hold for [Formula: see text]. As an application, the full Gromov–Witten theory of [Formula: see text] satisfies the Virasoro constraints through genus [Formula: see text].
In this paper, we show that a smooth bounded domain in [Formula: see text] admitting partial pseudoconvex exhaustion remains partial pseudoconvex. The main ingredient of the proof is based on a new characterization of hyper-[Formula: see text]-convex domains. Furthermore, we get several convex analogies in [Formula: see text].
In this paper, we introduce the generalized Nochka weights for the family of hyperplanes in the projective space [Formula: see text], defined via their generalized Nochka diagram and give an improvement of Cartan–Nochka’s theorem for holomorphic mappings from the complex plane into [Formula: see text] that intersect these hyperplanes.
In this paper, we solve Problem B raised recently by Zhai-Mo-Hu [Internat. J. Math. 36 (2025) 2550038]. As a direct consequence, we obtain a complete classification for all umbilic-free hypersurfaces of the (n + 1)-dimensional unit sphere Sn+1 (n >= 3) with semi-parallel Blaschke tensor.
Let G := K & ltimes; N be the semidirect product associated to the Gelfand pair (K,N), where N is a connected and simply connected nilpotent Lie group, and K is a compact subgroup of the automorphism group, Aut(N), of N. Under some assumptions on the pair (K,N), we give a precise connection between the spherical representations and the small representations of G, which are the irreducible representations of G that cannot be Hausdorff separated from the trivial one-dimensional representation 1(G )of G (called the cortex of G and denoted by cor(G)). More precisely, we show that the set of the small representations of G(cor(G)) is strictly contained in the closure of the set of the spherical representations of G (with respect to the Fell topology). Furthermore, we prove that any spherical representation of G cannot be a small representation. In the special case, (N is abelian) we show that the converse of this assertion is not true.
Let f is an element of S-k(SL2(Z)) be a normalized Hecke eigenform of integral weight k for the full modular group. In this paper, we study the average behaviour of Fourier coefficients of t-fold product L-function. More precisely, we establish the asymptotics of power moment associated to the sequence {lambda(f circle times f circle times & centerdot; & centerdot; & centerdot;circle times l) (f)(Q(x))}(Q is an element of SD) , (2)(x is an element of Z), where f circle times f circle times & centerdot; & centerdot; & centerdot; circle times(l) f denotes the l-fold product of f, and S-D denotes the set of inequivalent primitive integral positive-definite binary quadratic forms (reduced forms) of fixed discriminant D < 0. As a consequence, we prove results concerning the behavior of sign changes associated to the sequence {lambda(f circle times f circle times & centerdot; & centerdot; & centerdot;circle times l) (f)(Q(x))}(Q is an element of SD) , (2)(x is an element of Z) of Fourier coefficients of l-fold product L-function for an odd positive integer l.
This paper investigates the continuity mapping properties of local multilinear maximal commutator and its fractional variant on the first-order Sobolev spaces. The continuity of local multilinear maximal commutator and its fractional variant on the first-order Sobolev spaces are proved under the symbols in the first-order Sobolev spaces. The main results not only extend the main result of F. Liu, Q. Xue and K. Yabuta [Sobolev boundedness and continuity for commutators of the local Hardy-Littlewood maximal function, Ann. Fenn. Math. 47 (2022) 203-235] to a multilinear version and fractional version, but also essentially address some questions motivated by Liu and Xi [Sobolev regularity for commutators of the fractional maximal functions, Banach J. Math. Anal. 15 (2021) 1-36] and Zhang and Liu [Regularity for commutators of the local multilinear fractional maximal operators, Adv. Nonlinear Anal. 10 (2021) 849-876]. It should be pointed out that the Sobolev continuity result for the multilinear fractional maximal commutator is new even in the linear case m = 1.
The Morgan-type uncertainty principle and unique continuation properties for abstract Schrodinger equations with time-dependent operator potentials are obtained in H-valued function space. The equations contain abstract linear operators in abstract Hilbert spaces H. So, we can obtain unique continuation properties for different classes of Schrodinger-type equations by choosing the space H and linear operator function A(x), which occur in a wide variety of physical systems.
In this paper, we solve the complex Monge-Ampere equation for measures with a pluripolar part in the Cegrell class on compact Kahler manifolds, and require that the pluripolar part is solvable within this framework. This result generalizes the classical results obtained by Cegrell in bounded hyperconvex domains. We also discuss the properties of the complex Monge-Amp & egrave;re operator in some special cases.
We consider gaps that arise in the possible numbers of non-vanishing components for certain linear systems that arise when studying rational sphere maps. When the source dimension is n, we find an explicit formula for an integer G(n) such that every number exceeding G(n) is the value for the minimum embedding dimension of some polynomial (hence rational) sphere map. We conjecture, but do not prove, that G(n) is sharp. This number is asymptotic to root 2n(3/2); this asymptotic value improves what has been suggested to be quadratic in n. Our methods are combinatorial.
This paper classifies all left-invariant Hermitian and hyper-Hermitian structures on Lie groups whose commutator groups are one-dimensional. Furthermore, for each natural number n = 2k and n = 4k (k > 1 for the case of hyper-Hermitian structures), we construct an n-dimensional Lie group with a one-dimensional commutator group that possesses left-invariant Hermitian and hyper-Hermitian structures, respectively.
