In this work, we investigate the asymptotical behavior of orbits for non-Volterra quadratic stochastic operators depending on parameters and permutations. We find invariant sets, fixed and periodic points, and their type. We show that every orbit converges either to a fixed point or to a periodic orbit of period two; as a consequence, the operators are ergodic.
Let 𝒦_p(X) be the hyperspace of non-empty compact subsets of X that has to p∈ X . In this short note, we will show that if X is L-embedded in a compact space C, then 𝒦_p(X) is L-embedded in 𝒦_p(C) . This result implies that: 𝒦(𝔈^+) has dimension 1, where 𝔈^+ is the one-point-connected extension of Erdős space (see Theorem 1.1 in [14]). We give examples of spaces X, such that 𝒮_c(X) has dimension 1, answering Question 6.5 of [7], and we give a partial answer to Question 3.6 in [14].
Using the method of weighted metrics in the cone of the space of continuous functions, a global theorem on the existence and uniqueness of a non-negative solution for an integro-differential equation of the convolution type with a nonlinearly included derivative of the sought-for function is proved. It is shown that the solution can be found using the Picard-type method of successive approximations. Examples are provided to illustrate the obtained results.
We explore the lattice properties inherited by the space of Bochner integrable functions, denoted by L^1(μ ,E) , from a Banach lattice E. The aim of this work is to provide a generalization of classic convergence theorems for the Bochner integral. We will specifically focus on the monotone convergence theorem and Fatou’s lemma.
The paper is devoted to present considerations concerning the solvability on a bounded interval of a system of nonlinear integral equations, which create the generalization of a fractional-order neuron model under the electromagnetic field expressed in terms of Caputo fractional derivatives. Our approach depends on replacing the neuron models in question by a system of nonlinear integral equations of the Volterra–Stieltjes type. That enables us to apply suitable tools of nonlinear analysis and to obtain a transparent and easy-to-apply result. The established criteria significantly extend and refine the solvability theory for this class of generalized fractional neuron models.
We define matrix q-analogues of the higher-order Mersenne numbers via holomorphic functional calculus. For q∈ (0,1), k∈ℕ, and a complex square matrix A, let B_k,q(A):=[A^k]_q=(I-q^A^k)/(1-q) and set M^(k)_n,q(A)=∑ _j=0^n-1B_k,q(A)^j. All single-matrix identities are polynomial or holomorphic identities in B_k,q(A) and hence hold for arbitrary A, while commutativity is only required in genuinely multi-parameter matrix settings. We derive Binet-type formulas, matrix recurrences, a factorized Jackson q-exponential generating function, and a -embedding. A transfer-matrix method yields matrix q-Cassini, q-Catalan, and q-d’Ocagne identities, together with a determinantal Abel–Cassini identity for the nonautonomous system. We also prove spectral mapping results for σ (M^(k)_n,q(A) ) and include explicit 2× 2 examples, including a Jordan-block case. As q→ 1^-, the scalar specialization r=1 and A=2 recovers the higher-order Mersenne formulas of Prasad et al. [9].
The (α ,β ) -monogenic functions are defined as the solutions to the first-order differential equation α f 𝒟_x+ β𝒟_xf = 0, where α , β∈ℝ, α±β , and 𝒟_x denotes the Dirac operator in ℝ^m. In this paper, we address the question of whether a Hölder continuous function defined on a Lyapunov surface Γ⊂ℝ^m can be decomposed as the sum of the two boundary traces of a sectionally (α ,β ) -monogenic function. Our main tool consists of the Hardy projections associated with a singular integral operator arising in Clifford analysis, which turns out to be an involutive operator on the Hölder classes.
A ventana is a special kind of a basis of a topology, a ventana being a collection of sets which is closed wrt finite intersections and also wrt finite unions. Thus, a ventana is a generalization of a topology. Ventanas were the objects studied in a paper published in 2016. In 1973, T-categories were defined via axioms and were shown to be the sorts of categories in which initial sources and final sinks exist. The category Topology was, up to isomorphism, determined via axioms added to the T-category axioms. In this paper, it is established that Ventana is a T-category and is also, up to isomorphism, determined via axioms added to the T-category axioms.
Along the lines of Terai’s conjecture (Proc Jpn Acad Ser A Math Sci 70:22–26, 1994), we study the problem of determining the positive integer solutions of the purely exponential Diophantine equation a^x+b^y=c^z for fixed relatively prime positive integers a, b and c satisfying min{a, b,c}>1 and a+b=c^2 . Indeed, we prove that ( x, y,z) = (1, 1, 2) is the only positive integer solution of purely exponential Diophantine equation (n^2 - 3n + 1)^x+n^y=(n -1)^z for any positive integer n≥ 4 .
This study proposes a fractional-order smoking awareness model based on the Caputo derivative to capture the complex dynamics of smoking behavior and media-driven interventions. The model is analyzed to determine equilibrium points, compute the basic reproduction number ( R_0 ), and establish local and global stability of the smoking-free equilibrium. A semi-analytical solution is obtained using the homotopy perturbation general transform method (HPGTM). The accuracy of the proposed method is validated through comparison with the fractional Adams–Bashforth–Moulton (FABM) and fractional Euler methods. A convergence analysis is also performed, demonstrating the efficiency and rapid convergence of HPGTM. Numerical simulations illustrate the influence of the fractional-order parameter α and other key parameters on smoking dynamics. The results indicate that the proposed framework provides reliable insights for understanding smoking behavior and designing effective intervention strategies.
We study the existence of branched coverings between closed 3-manifolds, with emphasis on universal knots and links. We prove that the only closed 3-manifolds that admit a universal link are spherical. Furthermore, we distinguish between universal links and complement universal links and show that these notions do not coincide in general, by exhibiting infinitely many examples of complement universal links that are not universal. Also, we prove that there is no closed aspherical 3-manifold such that every closed, aspherical 3-manifold is a branched covering over it. Finally, we characterize the closed 3-manifolds admitting branching coverings from P^3 # P^3 , and deduce that there is no closed reducible 3-manifold, such that every closed reducible 3-manifold is a branched covering over it.
In this paper, we introduce a new regularity condition that characterizes the tameness of a composite singularity H=G∘ F in a sharp way. Our approach will provide a natural tool to study the topology of composite singularities H by relating the singular sets and Milnor sets of the component map germs F and G to those of H. We also study the invariance of tameness by ℒ -equivalence, ℛ -equivalence, and hence by 𝒜 -equivalence.
The braid alternation number of a knot K, denoted by Balt(K), is an invariant that measures how far a link is from being an alternating closed braid; it resembles the alternation number invariant. However, although some relationships with other invariants have been given, the value of this invariant is still unknown for some alternating knots with a few crossings. In this article, we expand upon the previously given estimation of the braid alternation number by calculating it for the case of most ten-crossing knots. We also provide some criteria for improving the tabulation.
A newly developed fuzzy structure known as a multidimensional fuzzy graph represents the most advanced fuzzy depiction of graphs, employing multidimensional fuzzy sets to illustrate and define data with uncertainty. A multidimensional fuzzy graph provides a framework that encompasses all necessary properties for depicting complex data. Nonetheless, being a hybrid model with infinite degrees of freedom, the operations in mdfg are particularly intricate. Consequently, the examination of mdfg with distinct characteristics such as vertex (edge) regularity and vertex (edge) irregularity enhances the feasibility of practical applications. This work primarily investigates regularity, irregularity, edge regularity, and its interconnections with operations including direct products, tensor products, unions, and others. The concluding section examines the prospective applications of regular multidimensional fuzzy graphs in relation to the connectivity index, offering appropriate demonstrations and contrasting them with earlier fuzzy models, including m-polar fuzzy sets.
We study a McKean–Vlasov forward–backward stochastic differential equation (FBSDE) in connection with the theory of stochastic differential mean-field games, particularly the weak (non-fully coupled) formulation described in Section 3.3.1 of [6]. Our main goal is to obtain regularity results for this McKean–Vlasov FBSDE, specifically classical and Malliavin differentiability.
We study several structural properties that determine when projections between homogeneous spaces give rise to equivariant fibrations. We focus on three key conditions: the projection G → G/H viewed as a conjugate H-fibration, the preservation of fibrations under the twisted-product functor G × _H(-) , and a straightening condition for G-homeomorphisms of the form X × I ≈ G × _H A . We prove that the first two conditions are equivalent, and that the third one provides a general mechanism for establishing the second one. This yields a unified framework for proving equivariant fibration theorems beyond the classical compact Lie group setting.
We provide sufficient conditions for the existence of periodic solutions and their stability of the 4-order differential equations ⃜u +(a_1u+a_0) ⃛u+(b_1u+1+b_0)ü+(c_1u+a_0)u̇+c_2u^2+b_0u=ε ^2F(t,u,u̇, ü, ⃛u,ε ), where a_0,a_1,b_0,b_1,c_1,c_2 are real parameters, ε is a small parameter and F is a C^2 2π -periodic function in the variable t. Moreover, we provide some applications.
In this paper, we present a overview of polynomial techniques in control theory providing a useful reference in this topics. We devote a special space to the usefulness of polynomials in the study of polynomial stability. We also explain how polynomials are an auxiliar tool to investigate some problems of controllability of systems, for example loop-closed control and motion planning problem.
In this paper, we investigate control problems related to a mathematical model that describes dynamic Coulomb’s frictional contact between a thermo-viscoelastic body and a thermally conductive rigid foundation. The contact conditions and thermal conductivity on the contact surface are modeled using Signorini’s boundary conditions. We formulate the problem’s weak form as a system combining variational and hemivariational inequalities. Our study establishes the existence and uniqueness of a weak solution to the model, and under certain additional assumptions, demonstrates the continuous dependence of the solution with respect to the problem data. Furthermore, we address a class of optimal control problems and establish the existence of optimal solutions.
