We study the irreducible representations of the Burnside groups B(m,3) . We relate their nontrivial images to 3 -periodic groups with center of order 3 and investigate such groups. We prove a result on the structure of these groups and use it to determine the dimensions of irreducible representations of B(5,3) and B(6,3) .
In this article, we have explored transcendental entire solutions of several nonlinear quadratic trinomial equations governed by the q -shift operator. While some of these equations involve the ordinary q -shift and the function itself, others incorporate combinations such as the q -shift together with a differential operator or the q -differential operator together with the function. Our findings are illustrated through several examples. These results refine and extend earlier work in a new direction, providing deeper insights into the nature of such equations and their transcendental entire solutions.
In this paper, within the framework of comparison by weights of polynomials, we establish necessary and sufficient conditions of hypoellipticity.
This paper is concentrated with the norm retrieval from sampled wavelet system measurements. More precisely, we show that the wavelet system {T_kba^jD_a^jψ}_j,k∈ℤ does norm retrieval on L_1(ℝ)∩ L_2(ℝ) , where a>1 , b>0 and ψ∈ L_2(ℝ) such that ψ̂∈ L_∞(ℝ) , lim_ξ→ 0ψ̂(ξ)∈ℂ∖{0} and ψ(a^i.)=λ_ijψ(a^j.) , for some λ_ij>0 , for all i,j∈ℕ . In particular, we show that if ψ∈ L_1(ℝ)∩ L_2(ℝ) such that ψ̂(0)≠ 0 and ψ(a^i.)=λ_ijψ(a^j.) , for some λ_ij>0 , for all i,j∈ℕ , then the wavelet system {T_kba^jD_a^jψ}_j,k∈ℤ does norm retrieval on L_2(ℝ) .
In the paper, the authors discuss the Schur m -power convexity of the ratio L_p(a,b)/M_r(a,b) between Lehmer’s mean L_p(a,b) and Hölder’s mean M_r(a,b) and obtain the Schur m -power convexity of Lehmer’s mean L_p(a,b) .
Bicomplex numbers are the generalization of complex numbers in four-dimensional settings. In this study, we derive a region containing the number of zeros of a bicomplex polynomial.
This paper mainly aims to study and explain meromorphic solutions of certain nonlinear partial differential equations from eikonal-type systems. We establish two theorems that provide conditions for the existence and structure of such solutions. Our results extend and refine earlier results, including a theorem by Li [1]. To illustrate the validity and usefulness of our results, we also present concrete examples.
This paper is concerned with the long-time behaviors of the complex Ginzburg–Landau equations with exponential nonlinearity. We first prove the well-posedness of solutions for the equations in an Orlicz space, and then, the existence of L^2(Ω) -global attractors are proved. Finally, we obtain the existence of global attractors in L^*_A(Ω) and H^1_0(Ω) .
This paper presents fundamental extensions of the classical Eneström–Kakeya theorem to the quaternionic setting, revealing new developments in the geometry of zeros for noncommutative polynomials. While the original theorem confines zeros of real polynomials with monotonic nonnegative coefficients to the unit disk, we establish broader quaternionic generalizations that relax these constraints and reveal richer zero structures. We characterize zero-containing regions for quaternionic polynomials with monotonicity imposed only on the first n-1 coefficients while allowing the leading coefficient f_n to vary freely in ℍ . We establish localization bounds sensitive to additive and multiplicative deviations from strict monotonicity, and reveal novel geometric zero structures such as 4D annuli and Cassini-type ovals arising uniquely from quaternionic noncommutativity. A computed example demonstrates applicability of our results to wider class of polynomials. This framework generalizes the classical Eneström–Kakeya theorem while highlighting uniquely quaternionic behaviors. To support our results, we also include low-dimensional visualizations that illustrate the geometric bounds for zeros of quaternionic polynomials.
In this paper, we mainly study the fractional elliptic equation: (a+b∫_ℝ^3|(-Δ)^s/2u|^2dx)(-Δ)^su+u=(|x|^-μ*|u|^p)|u|^p-2u, x∈ℝ^3, where μ∈(0,3) , s∈(0,1) , 2-μ/30 . For this equation, we will discuss it in two case. For s∈(0,3/4] , we prove the existence of solutions by establishing an equivalent system. For s∈(3/4,1) , we use the symmetric mountain pass lemma to prove that the equation has infinitely many solutions.
This article investigates the existence and explicit forms of solutions of a quadratic trinomial partial differential difference equation and a system of k th order partial differential difference equations in ℂ^2 . One main result extends a recent finding in the literature, while the other offers a novel contribution. We include several illustrative examples to demonstrate the accuracy and applicability of the results.
We provide complementary results for a family of models with dependence on their previous k -sum. Using a martingale-based approach, we establish a functional central limit theorem and analyze the limiting behavior of the center of mass. Additionally, we explore the connection between our findings and the study of certain reinforced random walks in the literature.
In this paper we formulate and solve a problem in metric fixed point theory and show that the results obtained herein are actual generalizations of certain previous results. The problem is formulated by combining three different existing lines of research. We discuss illustrative examples and include two applications wherein an integral equation and a third-order boundary value problem are solved by application of our results obtained herein. The examples demonstrate the fact that the class of functions to which the previous results are applicable are augmented by the results of the present paper.
This work proves the existence of an integrable function U(x) and a measurable set E⊂[0,1) that form a universal pair (𝐔,𝐄) in the sense of modification with respect to a multiplicative system.
In this work some uniqueness theorems for series with respect to the bounded Ciesielski system are proved. In particular, if the partial sums S_l(x)=∑_n=-k+2^la_nF_n(x) of the bounded Ciesielski series ∑_n=-k+2^∞a_nF_n(x) converge in measure to a bounded function f and sup_l|S_l(x)|<∞ when x∉ B , where B is some countable set, then this series is the Fourier series of the function f with respect to the bounded Ciesielski system.
The paper establishes the formulas for reconstructing a meromorphic function using its zeros, poles, and generalized kepstral coefficients, as well as using only its module on a circle.
Let k≥ 2 be a positive integer. Let ℱ be a family of meromorphic functions in D , whose zeros have multiplicity at least k+1 . Let h(≢0) be a holomorphic function in D , whose zeros are multiple. If each f∈ℱ and each z∈ D , f^(k)(z)≠ h(z) , then ℱ is normal at points for which h(z)=0 .
We investigate the count and placement of zeros of a generalized complex harmonic trinomial P_c(z). In particular, we prove that P_c(z) has (n+2k+j+1) number of zeros. Also, we obtain that sense-preserving and sense-reversing areas of P_c(z) lie in annular regions that enclose the critical curve of P_c(z) . At last we locate the zeros of P_c(z) and obtain annular sectors containing zeros of P_c(z). We demonstrate our findings with examples and figures.
