Let G be a simple algebraic group over an algebraically closed field, T a maximal torus, R the root system of (G, T), and h its Coxeter number. Let m be a positive integer smaller than h. Choose a base of R, whence a corresponding height function, and let R(m) be the set of roots whose height is a multiple of m. In a recent paper, S. Nadimpalli, S. Pattanayak and D. Prasad studied, for the purposes of character theory at torsion elements, the root systems R(m); in particular, they introduced a constant d_m which is always the dimension of a representation of the semisimple, simply-connected group with root system dual to R(m) and equals 1 if the roots of height m form a base of R(m), and they proved this property when R is of type A or C, and also in type B if m is odd. In this paper, we complete their analysis by determining a base of R(m) and computing the constant d_m in all cases. Further, we derive from our analysis the Kač coordinates of the standard principal embedding μ _m ↪ T when R is of classical type.
The present paper introduces and studies the constant, ρ = ∑ _n=1^∞1/∏ _k=1^n k!, defined as the infinite sum of the reciprocals of the superfactorials. It first establishes, by means of the ratio test, that the defining series of ρ converges absolutely and extremely rapidly, ensuring that the constant is well defined and numerically computable to arbitrary precision. It then proceeds to study the constant ρ by studying its nature and its connections to other mathematical functions. It also proves that the constant is an irrational number. The constant ρ thus enriches the family of known irrational constants and may possess further arithmetic properties worthy of investigation.
We prove that if U:c_0( 𝒳_1) ×···× c_0( 𝒳_n) → Y is strongly 1-summing, then all U∘( σ _k,… ,σ _k) are strongly 1-summing and ∑ _k=1^∞π _1^strong( U∘( σ _k, … ,σ _k) ) <∞ . We indicate some situations in which the converse is true and show that in general the converse is not true. We give, for various concrete examples of operators, the necessary and sufficient conditions for these to be strongly 1-summing.
Betti numbers serve as one of the most insightful and fruitful tools for revealing the correlation between the topology and geometry of Riemannian manifolds and the algebraic structure of the curvature tensor. This work deals with finding Betti numbers of compact Yamabe and generalized quasi-Yamabe gradient soliton. We prove that the Betti numbers of a compact oriented generalized quasi-Yamabe gradient soliton vanish and conclude that the manifold is homeomorphic to the n-sphere 𝕊^n in the simply connected case. Moreover, lower bounds of the diameter for compact Yamabe and quasi-Yamabe gradient soliton are derived.
In this article, we give condition on L^p -functions to form multipliers. Next, we produce similar results for product of two weighted L^p -functions. Thereafter, we derive new multipliers as convolution of two Sobolev space functions. After that, we study the L^p – L^q boundedness of the operators corresponding to a variant of Bochner–Riesz multiplier. We also give a simple proof of generalised Hörmander multiplier theorem.
Let ℍ_k^* denote the set of all primitive holomorphic cusp forms f of even integral weight k for the full modular group SL_2(ℤ) . Suppose that λ _f⊗ f ⊗ f(n) , λ _f⊗ f ⊗ g(n) and λ _sym^2f ⊗ f(n) are the n-th normalized Fourier coefficient of the the triple product L-functions L(s, f⊗ f ⊗ f) , L(s, f⊗ f ⊗ g) and the GL(3)× GL(2) L-function L(s, sym^2f ⊗ f) associated with two distinct cusp forms f ∈ℍ_k'^* and g∈ℍ_k”^* , respectively. In this paper, we establish quantitative results on the number of sign changes of the sequences {λ _f⊗ f ⊗ f(n)} , {λ _f⊗ f ⊗ g(n)} and {λ _sym^2f ⊗ f(n)} for n≥ 1 over short intervals, thereby improving upon previous results.
The crystal limit C(K_0) of the q-family of C^* -algebras C(K_q) was introduced by Giri and Pal for all K=SU(n+1), n≥ 2 . This article aims to prove that the crystal limit C(K_0) has the property that the representations of C(K_q) give rise to a representation of C(K_0) by sending generators of C(K_0) to the limit of (scaled) generators of C(K_q) and every representation of C(K_0) occurs in this way. This work addresses a question raised by Giri and Pal (J. Noncommut. Geom. 20 (2026), no. 2, 657–703). As a consequence, one can realize C(K_0) as the C^* -algebra generated by the limit operators of faithful representations of C(K_q) .
Katzman proved that the regularity of an edge ring of a graph is bounded below by its induced matching number. In this article, we study the homological invariants of edge rings of an infinite family of graphs with induced matching number one and regularity two. In particular, we study the structure of the minimal free resolution of the edge ring associated to the complement of the square of an n-cycle; 𝒞_n^2 , n>6 . We derive combinatorial formulae for computing all the graded Betti numbers of an edge ring of 𝒞_n^2 . Also, we deduce that the regularity and projective dimension of the edge ring of 𝒞_n^2 are 2 and n-2 , respectively. Further, we obtain the lower bound for the regularity of powers of edge ideals of the complement graph of second power of an n-cycle.
The concept of Davenport constant of a group G is one of the most celebrated and widely studied topics in zero-sum theory, a central area of Additive Combinatorics. It has many nice generalizations and remarkable applications in many branches of mathematics. The notion of the consecutive Davenport constant of a group G, introduced in [11] just a couple of years ago, is a recent addition to the literature. In this paper, we present a generalization of this notion and determine its exact value for any finite module over a commutative ring with identity. We also establish general lower and upper bounds for this invariant, specifying the cases where these bounds are sharp, and provide a complete characterization of all extremal sequences for this generalized consecutive Davenport constant.
This paper studies the vertex decomposability of two complexes derived from a finite simple graph G: the Stanley-Reisner complex of the 3-path ideal I_3(G) and the 3-path complex Δ _3(G) . For the first, we prove that the Stanley-Reisner complex of I_3(G) is vertex decomposable for a specified class of chordal graphs. For the second, a pure 2-dimensional complex, we establish two main results: (i) Δ _3(G) is vertex decomposable for every connected graph that contains no cycles of length five or greater, and (ii) vertex decomposability is preserved under the graph join operation, meaning Δ _3(G_1 ∨ G_2) is vertex decomposable for all graphs G_1 and G_2 .
We provide an infinite collection of exceptional quartic number fields by specifying exceptional units α of degree 4 by means of their minimal polynomials. Along with α , we show that α ^2 is also an exceptional unit. Our collection of fields is rich enough to have any real quadratic field as a subfield of one of them. On the other hand, another infinite collection of quartic exceptional fields with no quadratic subfields is also provided. In both the collections, the fields have unit rank 3, they are non-Galois extensions of Q , and their normal closures have Galois groups D_4 and S_4 , respectively. We further show that one can find a vast collection of higher degree exceptional number fields as an easy consequence of Perron’s irreducibility theorem.
As is well known, generalized normalities and supplementarities of all maximal subgroups of a Sylow p-subgroup P are closely related to the structure of a finite group. Viewing from the point of "all", we construct a set F(P,G) = {P-1 < & centerdot;P|P boolean AND O-p(G) not less than P-1 } which consists of "some" maximal subgroups of P instead of "all". Further, we investigate the influence of nearly SS-embedded and SS-supplemented properties of its elements on the p-supersolvability of a group. To some extent, our results also improved some Theorems.
We develop a method to derive the first term identities for Eisenstein series, inspired by Langlands' ideas on the spectral decomposition of automorphic forms, specifically for the exceptional group G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{2}$$\end{document}.
In this paper, we study non-reflexive Banach spaces X for which the quotient space X^**/X is reflexive. Such spaces were first introduced by James R Clark [Proc. Amer. Math. Soc. 36 (1972), pp. 421–427] where they were called coreflexive spaces. In Theorem 5, we show that a space X is coreflexive if and only if every separable subspace Y⊆ X is coreflexive, provided that X is w ^* -sequently dense in its bidual X^** . We show that coreflexive spaces are stable under ℓ ^p -sum for 1
In this article, We prove the following results: (1). We show that if R is an affine C-algebra of dimension 4, where C is either a subfield F of 𝔽_p or C=ℤ . Then WMS_3(R) is an abelian group. (2). Let R be a commutative ring with sdim( R)≤ 3, n≥ 3 . We show that if [v]/[w] is the quotient of orbits [v], [w]∈Um_n(R)/E_n(R) , then [w]/[v] is the inverse of [v]/[w].
In this paper, we estimate the minimal cardinality of the transcendental dilated sumset S+μ· T , where S and T are non-empty finite subsets of ℝ -Module ℝ and μ is a transcendental number. Additionally, we optimize the lower bounds for the cardinalities of the sets S+NT and S+N /̂p̂ĥân̂t̂ôm̂î T , where N, S, and T are finite subsets of positive elements of ℤ -Module ℤ .
