Abstract We study foliations ℱ {\mathscr{F}} on projective complete intersection K3 surfaces X ↪ ℙ n {X\hookrightarrow\mathbb{P}^{n}} , where ℱ {\mathscr{F}} has isolated singularities and it is the restriction of a foliation of degree d on ℙ n {\mathbb{P}^{n}} that leaves X invariant. We compute the values of the degrees d for which ℱ {\mathscr{F}} is uniquely determined by its singular scheme.
A primary part in the rational homology cobordism group is generated by manifolds for which the order of the first homology is a power of a given prime. Kim and Livingston proved that the rational homology cobordism group does not equal the sum of primary parts. The current paper defines composite parts, generated by manifolds for which the order of the first homology is a product of powers of primes in a given set. We show that the rational homology cobordism group is not generated by composite parts with bounded number of distinct prime factors. Moreover, we give a necessary and sufficient condition for two sums of composite parts to be equal. We also prove the rational homology cobordism group is not right primary decomposable, answering a question of Cha.
A necessary criterion for injective solutions to the set-theoretical Yang-Baxter equation in terms of q-cycle sets is given, improving Soloviev's syntactic criteria. Supporting evidence is provided that the new criterion is sufficient. Generalizing augmented rack solutions, q-cycle groups are revisited, and a general extension theorem for the underlying solution is established. Close relationships between injective solutions, generic coverings, and extensions of q-cycle groups, are obtained. Several characterizations of generic coverings are given.
Very recently, two new notions of para-linear mappings and weak associative orthonormal bases were introduced in octonionic functional analysis, which have been proved to be powerful in formulating the basic theory, such as the Riesz representation theorem and the Parseval theorem. In this article, we continue exploring more properties of these two concepts and initiate the study of octonionic para-linear isometric operators. Surprisingly, it is proven that the condition of the para-linear operator on a Hilbert octonionic bimodule being an isometric isomorphism is equivalent to it mapping any associative orthonormal basis to a weak associative orthonormal basis, which implies also that an octonionic matrix is an isometry if and only if the system of its row vectors is a weak associative orthonormal basis. Furthermore, we introduce the concept of para-linear partial isometric operators and establish the aforementioned analogue in this new setting. Based on these facts, we can provide naturally a new viewpoint of James questions by modifying the definition of octonionic Stiefel space.
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 C * {C<^>{*}} -algebras of pseudoquonic operators leveraging on different methods of functional analysis and operator theory.
We obtain an asymptotic formula for all moments of Dirichlet L-functions L(1, chi) modulo p when averaged over a subgroup of characters chi of size p-1/d with phi(d) = o(log p). Assuming the infinitude of Mersenne primes, the range of our result is optimal and improves and generalises the previous result of S. Louboutin and M. Munsch (2022) for second moments. We also use our ideas to get an asymptotic formula for the second moment of L(1/2, chi) over subgroups of characters of similar size. This leads to non-vanishing results in this family where the proportion obtained depends on the height of the smallest rational number lying in the dual group. This improves a recent result of this type due to & Eacute;. Fouvry, E. Kowalski and Ph. Michel (2024). Additionally, we prove that, in both cases, we can take much smaller subgroups for almost all primes p.
For a complex number x, parallel to x parallel to := min { | x - m | : m is an element of & Zopf; }. Let k >= 1 be an integer, and let K be a number field. Let alpha 1 , & mldr; , alpha k be algebraic numbers with | alpha i | >= 1 and let d i {d_{i}} denotes the degree of alpha i for 1 <= i <= k. Set d = d 1 + & ctdot; + d k. In this article, we show that if the inequality 0 < parallel to lambda 1 q alpha 1 n + & ctdot; + lambda k q alpha k n parallel to < theta n q d + epsilon has infinitely many solutions in ( n , q , lambda 1 , & mldr; , lambda k ) is an element of & Nopf; 2 x ( K x ) k with absolute logarithmic Weil height of lambda i is small compared to n and some theta is an element of ( 0 , 1 ), then, in particular, the tuple ( lambda 1 q alpha 1 n , & mldr; , lambda k q alpha k n ) is pseudo-Pisot, and at least one of alpha i {\alpha_{i}} is an algebraic integer. This result can be viewed as Roth-type theorem for linear combinations of powers of algebraic numbers over & Qopf; . The case q = 1 {q=1} was recently proved in [A. Kulkarni, N. M. Mavraki and K. D. Nguyen, Algebraic approximations to linear combinations of powers: An extension of results by Mahler and Corvaja-Zannier, Trans. Amer. Math. Soc. 371 2019, 6, 3787-3804], which is a generalization of Mahler's question proved in [P. Corvaja and U. Zannier, On the rational approximations to the powers of an algebraic number: Solution of two problems of Mahler and Mend & egrave;s France, Acta Math. 193 2004, 2, 175-191]. As a consequence of our result, we obtain the following generalization of this question: let alpha > 1 be an algebraic number with d = [ & Qopf; ( alpha ) : & Qopf; ]. For a given epsilon > 0, if the inequality0 < parallel to lambda q alpha n parallel to < theta n q d + epsilon 0 has infinitely many solutions in the tuples ( n , q , lambda ) is an element of & Nopf; 2 x K x with absolute logarithmic Weil height of lambda is small compared to n and theta is an element of ( 0 , 1 ) {\theta\in(0,1)} , then some power of alpha is a Pisot number. As an application of this result, we deduce the transcendence of certain infinite products of algebraic numbers.
By making use of a very-well-poised (6)phi(5) summation, the creative microscoping method, and the Chinese remainder theorem for coprime polynomials, we establish some new q-supercongruences, including some Dwork-type q-supercongruences. As a consequence, we obtain the following result: for any prime p = 1 ( mod 4) and integer r >= 1, Sigma(pr-1/2)(k=0) (8k + 1) (1/4)(k)(3) (1/2)k/k!(3)(3/4)k = p(r) ( mod p(r+3)), where (x)(0) = 1 and (x)(n) = x(x + 1) center dot center dot center dot (x + n - 1) for n >= 1.
In this paper, we establish a necessary and sufficient condition for a bounded linear operator on the Bergman space to be a slant Toeplitz operator having an analytic symbol. Moreover, we obtain a complete characterization for the commutativity of two slant Toeplitz operators with symbols p + phi {\overline{p}+\varphi} and psi on the Bergman space, where phi and psi are two bounded analytic functions, and p is an analytic polynomial.
Let mu denote the infinite convolution generated by {(N-k, B-k)}(infinity)(k=1) given by mu = delta(N1)-1(B1) * delta(N1N2)-B-1(2) * center dot center dot center dot * delta(N1N2...N-k)B--1(k) * center dot center dot center dot, where B-k is a complete residue system for each integer k > 0. We write nu(>k) = delta(N)-1 B-k+1(k+1) * delta((Nk+1Nk+2))-1B(k+2) * center dot center dot center dot. Since the elements in Bk may have very large absolute values, the infinite convolution may not be compactly supported. In this paper, we study the necessary and sufficient conditions for such infinite convolutions being spectral measures. The necessary conditions for the spectrality of mu mainly depend on the properties of the polynomials generated by the complete residue systems. The main result shows that if every B-k satisfies uniformly discrete zero condition, and if {nu(>k)}(infinity)(k=1) is tight, then #B-k|N-k for all integers k >= 2. For some special complete residue systems, we provide the necessary and sufficient conditions for mu being a spectral measure.
In this paper, we study truncated Hankel operators with matrix-valued symbols which are compressions of Hankel operators on the model space corresponding to a square inner matrix. An operator equation characterization of such operators is given and some basic questions such as which symbols give rise to a zero truncated Hankel operator is answered. Additionally, we show that truncated Hankel operators with matrix-valued symbols exhibit a specific type of shift symmetry. We also note that by considering truncated Hankel operators with matrix-valued symbols, we can give a unified treatment of asymmetric truncated Hankel operators.
Let G be an affine or hyperbolic rank 2 Kac-Moody group over a finite field F-q. Let X = Xq+1 be the Tits building of G, the (q + 1)-homogeneous tree, and let Gamma be a non-uniform lattice in G. When Gamma is a standard parabolic subgroup for the negative BN-pair, we define Eisenstein series on Gamma\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.
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} .
Let k,j and n be positive integers such that k is odd, and bothj and n are even, satisfyingj equivalent to n mod 4. Let f and g be primitive forms of weight 2k +j-2 and k + j/2-n/2-1, respectively, for SL2(Z). Then we propose a conjecture on the congruence between the Klingen-Eisenstein lift of the Miyawaki lift off and g of type II and a certain lift of a vector-valued Hecke eigenform of weight (k + j, k) for Sp(2)(Z). This conjecture implies Harder's conjecture. Through this formulation, we prove Harder's conjecture in some cases.
In this paper, we give explicit error bounds for the asymptotic expansion of the shifted distinct partition function q(n + s) for any nonnegative integer s. Then based on this refined asymptotic formula, we give the exact thresholds of n for the inequalities derived from the invariants of the quartic binary form, the double Tur & aacute;n inequalities, the Laguerre inequalities and their corresponding companion versions.
In this article we study the stability of Kernel sheaf obtained from a generating subspace of rank one torsion-free sheaf on an integral nodal curve.
An effective upper bound is established for the least non-trivial integer solution to the system of cubic forms(1) { F = c 1 x 1 3 + c 2 x 2 3 + & ctdot; + c n x n 3 = 0 , G = d 1 x 1 3 + d 2 x 2 3 + & ctdot; + d n x n 3 = 0 , under the "M-good" condition for n >= 16, where c 1 , & mldr; , c n and d 1 , & mldr; , d n are integers. Additionally, a range is derived for the probability that randomly selected simultaneous equations satisfy the M-good condition.
Given q is an element of & Nopf;(>= 3) and a finite set A subset of & Qopf;, let K(q, A) = { (infinity) & sum;( i=1) ai/q i : ai is an element of A for all i is an element of & Nopf;}. For p is an element of & Nopf;(>= 2) let D-p subset of & Ropf; be the set of all rational numbers having a finite p-ary expansion. We show in this paper that for p is an element of & Nopf;(>= 2) with gcd(p, q) = 1, the intersection D-p boolean AND K(q, A) is a finite set if and only if dim(H) K(q, A) < 1, which is also equivalent to the fact that the set K(q, A) has empty interior. We apply this result to study the spectral eigenvalue problem. For a Borel probability measure mu on & Ropf;, a real number t is an element of & Ropf; is called a spectral eigenvalue of mu if both E(Lambda) = {e(2 pi i lambda x) : lambda is an element of Lambda} and E(t Lambda) = {e (2 pi it lambda x) : lambda is an element of Lambda} are orthonormal bases in L-2 (mu) for some Lambda subset of & Ropf;. For any self-similar spectral measure generated by a Hadamard triple, we provide a class of spectral eigenvalues which is dense in [0, +infinity), and show that every eigen-subspace associated with these spectral eigenvalues is infinite.
