Main Theorem Research Articles

Simultaneous Solution of a Fuzzy Global Optimization Problem Using a Fixed Point Method

In this paper, we adopt a fixed point approach towards a problem of finding fuzzy distance between two subsets of a fuzzy metric space. This is accomplished simultaneously by determining two different pairs of points which are obtained by finding an optimal approximate solution of a coupled fixed point equation for an appropriate coupled contractive operator. By its nature, the problem is a global optimization problem. The coupled contraction defined here has certain properties suitable for applications to the optimization problem considered in this paper. The main result is illustrated with an example. By applications of the main theorem, a new coupled fixed point result is derived in fuzzy metric space and a new coupled proximity point result is obtained in metric spaces. This work is a continuation of a newly emerged line of research on global optimization in the context of fuzzy metric spaces.

A second main theorem of finite ramified coverings

In this paper, we study a second main theorem for holomorphic curves from finite ramified coverings of the complex line to complex projective varieties intersecting hypersurfaces in subgeneral position.

Critical Analyzing on Some New Application of Almost Decreasing Sequence to Legendre Series Associated with [B] Sum

In this study, we present a novel application of a nearly decreasing sequence connected to [B] summation. Additionally, a novel and well-known arbitrary result was attained by applying the main theorem. The current findings are validated by taking into account the paper’s circumstance of a prior result.

Bers’ simultaneous uniformization and the intersection of Poincaré holonomy varieties

We consider the space of ordered pairs of distinct {mathbb{C}{mathrm{P}}}^{1}-structures on Riemann surfaces (of any orientations) which have identical holonomy, so that the quasi-Fuchsian space is identified with a connected component of this space. This space holomorphically maps to the product of the Teichmüller spaces minus its diagonal.In this paper, we prove that this mapping is a complete local branched covering map. As a corollary, we reprove Bers’ simultaneous uniformization theorem without any quasi-conformal deformation theory. Our main theorem is that the intersection of arbitrary two Poincaré holonomy varieties (operatorname{SL}_{2}mathbb{C}-opers) is a non-empty discrete set, which is closely related to the mapping.

Traced Monads and Hopf Monads

A traced monad is a monad on a traced symmetric monoidal category that lifts the traced symmetric monoidal structure to its Eilenberg-Moore category. A long-standing question has been to provide a characterization of traced monads without explicitly mentioning the Eilenberg-Moore category. On the other hand, a symmetric Hopf monad is a symmetric bimonad whose fusion operators are invertible. For compact closed categories, symmetric Hopf monads are precisely the kind of monads that lift the compact closed structure to their Eilenberg-Moore categories. Since compact closed categories and traced symmetric monoidal categories are closely related, it is a natural question to ask what is the relationship between Hopf monads and traced monads. In this paper, we introduce trace-coherent Hopf monads on traced monoidal categories, which can be characterized without mentioning the Eilenberg-Moore category. The main theorem of this paper is that a symmetric Hopf monad is a traced monad if and only if it is a trace-coherent Hopf monad. We provide many examples of trace-coherent Hopf monads, such as those induced by cocommutative Hopf algebras or any symmetric Hopf monad on a compact closed category. We also explain how for traced Cartesian monoidal categories, trace-coherent Hopf monads can be expressed using the Conway operator, while for traced coCartesian monoidal categories, any trace-coherent Hopf monad is an idempotent monad. We also provide separating examples of traced monads that are not Hopf monads, as well as symmetric Hopf monads that are not trace-coherent.

Periodic Solution Problems for a Class of Hebbian-Type Networks with Time-Varying Delays

By using Gronwall’s inequality and coincidence degree theory, the sufficient conditions of the globally exponential stability and existence are given for a Hebbian-type network with time-varying delays. The periodic behavior phenomenon is one of the hot topics in network systems research, from which we can discover the symmetric characteristics of certain neurons. The main theorems in the present paper are illustrated using a numerical example.

A note on $\mathcal{M}$-normal embedded subgroups of finite groups

In this note, we obtain a criterion for $p$-nilpotency of a finite group. We not only simplify the proofs of the main theorems of Chen et al. [On $\mathcal{M}$-normal embedded subgroups and the structure of finite groups, Math. Scand. 127(2021), no. 2, 243–251] but also generalize these results.

The only limit cycle that appears in damped harmonic oscillators affected by state-dependent impulses

This study is devoted to planar hybrid dynamical systems with state-dependent impulses to investigate phenomena related to sudden changes in the velocity of an object. In particular, the effect of the state-dependent impulses on the appearance of limit cycles is addressed, which has rarely been studied to date. When a toddler is placed on a swing and pushed gently from behind, the discontinuous changes in the direction and speed of movement of both the toddler and the swing can be described using such a hybrid dynamical system. The main theorem presented in this study reveals that, regardless of the amount of state-dependent impulses, they yield only one stable limit cycle within the region that is identified by the considered system. Moreover, state-dependent impulses lead to destabilization of the equilibrium point. In the proof of the main theorem, a Poincaré mapping is derived by examining the behavior of positive semi-orbits, and the characteristics of the sequence that is obtained from the Poincaré mapping are investigated in detail. Several simulations are performed to demonstrate the main theorem.

Constructive aspects of Riemann’s permutation theorem for series

Abstract The notions of permutable and weak-permutable convergence of a series $\sum _{n=1}^{\infty }a_{n}$ of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s principle BD- $\mathbb {N}$ implies that every permutably convergent series is absolutely convergent. Since there are models of constructive mathematics in which the Riemann permutation theorem for series holds but BD- $\mathbb{N}$ does not, the best we can hope for as a partial converse to our first theorem is that the absolute convergence of series with a permutability property classically equivalent to that of Riemann implies BD- $\mathbb {N}$ . We show that this is the case when the property is weak-permutable convergence.

Inequalities concerning maximum modulus and zeros of random entire functions

AbstractLet $f(z)=\sum\limits_{j=0}^{\infty} a_j z^j$ be a transcendental entire function and let $f_\omega(z)=\sum\limits_{j=0}^{\infty}\chi_j(\omega) a_j z^j$ be a random entire function, where $\chi_j(\omega)$ are independent and identically distributed random variables defined on a probability space $(\Omega, \mathcal{F}, \mu)$. In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher and Steinhaus entire functions. We prove that, for almost all functions in the family and for any constant C > 1, there exist a constant $r_0=r_0(\omega)$ and a set $E\subset [e, \infty)$ of finite logarithmic measure such that, for $r \gt r_0$ and $r\notin E$, \begin{equation*} |\log M(r, f)- N(r,0, f_\omega)|\le (C/A)^{\frac1{B}}\,\log^{\frac1{B}}\,\log M(r,f) +\log\,\log M(r, f), \qquad a.s. \end{equation*}where $A, B$ are constants, $M(r, f)$ is the maximum modulus and $N(r, 0, f)$ is the integrated zero-counting function of f. As a by-product of our main results, we prove Nevanlinna’s second main theorem for random entire functions. Thus, the characteristic function of almost all functions in the family is bounded above by an integrated counting function, rather than by two integrated counting functions as in the classical Nevanlinna theory. For instance, we show that, for almost all Gaussian entire functions fω and for any ϵ > 0, there is r0 such that, for $r \gt r_0$, \begin{equation*} T(r, f) \le N(r,0, f_\omega)+\left(\tfrac12+\epsilon\right) \log T(r, f). \end{equation*}

The Stability and Well-Posedness of Fixed Points for Relation-Theoretic Multi-Valued Maps

The purpose of this study is to present fixed-point results for Suzuki-type multi-valued maps using relation theory. We examine a range of implications that arise from our primary discovery. Furthermore, we present two substantial cases that illustrate the importance of our main theorem. In addition, we examine the stability of fixed-point sets for multi-valued maps and the concept of well-posedness. We present an application to a specific functional equation which arises in dynamic programming.

Approximation of sequential fractional systems of Liouville–Caputo type by discrete delta difference operators

We present the Liouville–Caputo fractional difference method for the numerical evolution of the sequential differential equation of fractional order. Meanwhile, some binomial coefficients are considered in discrete fractional calculus to find and determine the corresponding sequence of continuous fractional order equations. Finally, a standard numerical test is offered in detail to demonstrate the validity of the main theorem.

A structure theory for stable codimension 1 integral varifolds with applications to area minimising hypersurfaces mod 𝑝

For any Q ∈ { 3 2 , 2 , 5 2 , 3 , … } Q\in \{\frac {3}{2},2,\frac {5}{2},3,\dotsc \} , we establish a structure theory for the class S Q \mathcal {S}_Q of stable codimension 1 stationary integral varifolds admitting no classical singularities of density > Q >Q . This theory comprises three main theorems which describe the nature of a varifold V ∈ S Q V\in \mathcal {S}_Q when: (i) V V is close to a flat disk of multiplicity Q Q (for integer Q Q ); (ii) V V is close to a flat disk of integer multiplicity > Q >Q ; and (iii) V V is close to a stationary cone with vertex density Q Q and supports the union of 3 or more half-hyperplanes meeting along a common axis. The main new result concerns (i) and gives in particular a description of V ∈ S Q V\in \mathcal {S}_Q near branch points of density Q Q . Results concerning (ii) and (iii) directly follow from parts of previous work of the second author [Ann. of Math. (2) 179 (2014), pp. 843–1007]. These three theorems, taken with Q = p / 2 Q=p/2 , are readily applicable to codimension 1 rectifiable area minimising currents mod p p for any integer p ≥ 2 p\geq 2 , establishing local structure properties of such a current T T as consequences of little, readily checked, information. Specifically, applying case (i) it follows that, for even p p , if T T has one tangent cone at an interior point y y equal to an (oriented) hyperplane P P of multiplicity p / 2 p/2 , then P P is the unique tangent cone at y y , and T T near y y is given by the graph of a p 2 \frac {p}{2} -valued function with C 1 , α C^{1,\alpha } regularity in a certain generalised sense. This settles a basic remaining open question in the study of the local structure of such currents near points with planar tangent cones, extending the cases p = 2 p=2 and p = 4 p=4 of the result which have been known since the 1970’s from the De Giorgi–Allard regularity theory [Ann. of Math. (2) 95 (1972), pp. 417–491] [Frontiere orientate di misura minima, Editrice Tecnico Scientifica, Pisa, 1961] and the structure theory of White [Invent. Math. 53 (1979), pp. 45–58] respectively. If P P has multiplicity > p / 2 > p/2 (for p p even or odd), it follows from case (ii) that T T is smoothly embedded near y y , recovering a second well-known theorem of White [Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1986, pp. 413–427]. Finally, the main structure results obtained recently by De Lellis–Hirsch–Marchese–Spolaor–Stuvard [arXiv:2105.08135, 2021] for such currents T T all follow from case (iii).

Plato’s Timaeus and optimal pentatonic scales

After a short review of Pythagorean theory of harmonic ratios and musical scales as it is described in Plato’s Timaeus treatise, the concept of ‘optimality of a sequence of (real) numbers with respect to Pythagorean ratios’ is defined and main theorem of this article proves that there are only three optimal sequences of length 6, which correspond to three well-known pentatonic scales which are used in many musical traditions (including Chinese, Japanese and others). It is also noted that a definition similar to our optimal scales has appeared in a treatise by Sadi-al-Din Urmavi, a thirteenth century Iranian musicologist.

Enlargements of Quantales

Abstract In this paper, we study the enlargements of quantales. We prove three main results. First, if Q is a factorizable quantale, then any matrix quantale over Q is an enlargement of Q; second, any unital Rees matrix quantale over a quantale Q with an identity is an enlargement of Q; third, two quantales are Morita equivalent if and only if they have a joint enlargement. To prove these theorems, we use quantale matrices and modules and Morita contexts of quantales. Our main theorems and their proofs are parallel to those known for idempotent rings.

Coefficient Problems of Quasi-Convex Mappings of Type B on the Unit Ball in Complex Banach Spaces

ABSTRACT In this paper, the sharp solutions of Fekete-Szegö problems are provided for class of quasi-convex mappings f 1 of type B and class of quasi-convex mappings f 2 of type B and order α defined on the unit ball in a complex Banach space, respectively, where x = 0 is a zero of order k + 1 of fi (x) − x (i = 1, 2). Compare with some recent works, our main theorems hold without additional restrictive conditions. Also, the proof of our main theorems are more simple than those given in the previous results.

Equationally defined classes of semigroups

We apply, in the context of semigroups, the main theorem from the authors’ paper “Algebras defined by equations” (Higgins and Jackson in J Algebra 555:131–156, 2020) that an elementary class {mathscr {C}} of algebras which is closed under the taking of direct products and homomorphic images is defined by systems of equations. We prove a dual to the Birkhoff theorem in that if the class is also closed under the taking of containing semigroups, some basis of equations of {mathscr {C}} is free of the forall quantifier. We also observe the decidability of the class of equation systems satisfied by semigroups, via a link to systems of rationally constrained equations on free semigroups. Examples are given of EHP-classes for which neither (forall cdots )(exists cdots ) equation systems nor (exists cdots )(forall cdots ) systems suffice.

Homogenization of Attractors to Ginzburg-Landau Equations in Media with Locally Periodic Obstacles: Critical Case

In this paper the Ginzburg-Landau equation is considered in locally periodic porous medium, with rapidly oscillating terms in the equation and boundary conditions. It is proved that the trajectory attractors of this equation converge in a weak sense to the trajectory attractors of the limit Ginzburg-Landau equation with an additional potential term. For this aim we use an approach from the papers and monographs of V.V. Chepyzhov and M.I. Vishik concerning trajectory attractors of evolution equations. Also we apply homogenization methods appeared at the end of the XX-th century. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) equation and prove the existence of trajectory attractors for this equation. Then we formulate the main theorem and prove it with the help of axillary lemmas.

On the function approximation by trigonometric polynomials and the properties of families of function classes over harmonic intervals

The article is devoted to research on approximation theory. When approximating functions by trigonometric polynomials, the spectrum is chosen from various sets. In this paper, the spectrum consists of harmonic intervals. Devices, various processes, perception of the senses have a limited range. In the mathematical modeling of numerous practical problems and in the further study of such mathematical models, it is sufficient to find a solution in this range. It is possible to study such models to some extent with the help of harmonic intervals. To prove the main theorem, an auxiliary lemma was proved, and elements of the theory of approximations with respect to harmonic intervals were used. For the constructed families of function classes associated with the best approximations by trigonometric polynomials with a spectrum of harmonic intervals, their relationship with classical Besov spaces is shown.

F-Contractions Endowed with Mann’s Iterative Scheme in Convex Gb-Metric Spaces

Recently, Ji et al. established certain fixed-point results using Mann’s iterative scheme tailored to Gb-metric spaces. Stimulated by the notion of the F-contraction introduced by Wardoski, the contraction condition of Ji et al. was generalized in this research. Several fixed-point results with Mann’s iterative scheme endowed with F-contractions in Gb-metric spaces were proven. One non-trivial example was elaborated to support the main theorem. Moreover, for application purposes, the existence of the solution to an integral equation is provided by using the axioms of the proven result. The obtained results are generalizations of several existing results in the literature. Furthermore, the results of Ji. et al. are the special case of theorems provided in the present research.