Linking Theorem Research Articles

A prime decomposition theorem for string links in a thickened surface

We prove a prime decomposition theorem for string links in a thickened surface. Namely, we prove that any non-braid string link [Formula: see text], where [Formula: see text] is a compact orientable (not necessarily closed) surface other than [Formula: see text], can be written in the form [Formula: see text], where [Formula: see text] is prime string link defined up to braid equivalence, and the decomposition is unique up to possibly permuting the order of factors in its right-hand side.

Prescribed energy saddle-point solutions of nonlinear indefinite problems

A minimax variational method for finding mountain pass-type solutions with prescribed energy levels is introduced. The method is based on application of the Linking Theorem to the energy-level nonlinear Rayleigh quotients which critical points correspond to the solutions of the equation with prescribed energy. An application of the method to nonlinear indefinite elliptic problems with nonlinearities that does not satisfy the Ambrosetti-Rabinowitz growth conditions is also presented.

Multiple subharmonic solutions in Hamiltonian system with symmetries

In this paper, we prove the existence of infinitely many non-constant geometrically distinct symmetric subharmonic solutions and symmetric subharmonic brake orbits for symmetric Hamiltonian systems which are superquadratic at zero and infinity by combining iteration theory of Maslov-type indices under various boundary conditions, the Galerkin approximation arguments and link theorem of critical point theory. Furthermore, we prove the existence of multiple geometrically distinct symmetric subharmonic solutions and symmetric subharmonic brake orbits for the symmetric Hamiltonian systems.

Pseudo links and singular links in the Solid Torus

In this paper we introduce and study the theories of pseudo links and singular links in the Solid Torus, ST. Pseudo links are links with some missing crossing information that naturally generalize the notion of knot diagrams, and that have potential use in molecular biology, while singular links are links that contain a finite number of self-intersections. We consider pseudo links and singular links in ST and we set up the appropriate topological theory in order to construct invariants for these types of links in ST. In particular, we formulate and prove the analogue of the Alexander theorem for pseudo links and for singular links in ST. We then introduce the mixed pseudo braid monoid and the mixed singular braid monoid, with the use of which, we formulate and prove the analogue of the Markov theorem for pseudo links and for singular links in ST. \smallbreak Moreover, we introduce the pseudo Hecke algebra of type A, $P\mathcal{H}_n$, the cyclotomic and generalized pseudo Hecke algebras of type B, $P\mathcal{H}_{1, n}$, and discuss how the pseudo braid monoid (cor. the mixed pseudo braid monoid) can be represented by $P\mathcal{H}_{n}$ (cor. by $P\mathcal{H}_{1, n}$). This is the first step toward the construction of HOMFLYPT-type invariants for pseudo links in $S^3$ and in ST. We also introduce the cyclotomic and generalized singular Hecke algebras of type B, $S\mathcal{H}_{1, n}$, and we present two sets that we conjecture that they form linear bases for $S\mathcal{H}_{1, n}$. Finally, we generalize the bracket polynomial for pseudo links in ST.

Link theorem and distributions of solutions to uncertain Liouville-Caputo difference equations

<p style='text-indent:20px;'>We consider a class of initial fractional Liouville-Caputo difference equations (IFLCDEs) and its corresponding initial uncertain fractional Liouville-Caputo difference equations (IUFLCDEs). Next, we make comparisons between two unique solutions of the IFLCDEs by deriving an important theorem, namely the main theorem. Besides, we make comparisons between IUFLCDEs and their <inline-formula><tex-math id="M1">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths by deriving another important theorem, namely the link theorem, which is obtained by the help of the main theorem. We consider a special case of the IUFLCDEs and its solution involving the discrete Mittag-Leffler. Also, we present the solution of its <inline-formula><tex-math id="M2">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths via the solution of the special linear IUFLCDE. Furthermore, we derive the uniqueness of IUFLCDEs. Finally, we present some test examples of IUFLCDEs by using the uniqueness theorem and the link theorem to find a relation between the solutions for the IUFLCDEs of symmetrical uncertain variables and their <inline-formula><tex-math id="M3">\begin{document}$ \varrho $\end{document}</tex-math></inline-formula>-paths.</p>

Satellite constructions and geometric classification of Brunnian links

We study satellite operations on Brunnian links. First, we find two special satellite operations, both of which can construct infinitely many distinct Brunnian links from almost every Brunnian link. Second, we give a geometric classification theorem for Brunnian links, characterize the companionship graph defined by Budney in [JSJ-decompositions of knot and link complements in [Formula: see text], Enseign. Math. 3 (2005) 319–359], and develop a canonical geometric decomposition, which is simpler than JSJ-decomposition, for Brunnian links. The building blocks of Brunnian links then turn out to be Hopf [Formula: see text]-links, hyperbolic Brunnian links, and hyperbolic Brunnian links in unlink-complements. Third, we define an operation to reduce a Brunnian link in an unlink-complement into a new Brunnian link in [Formula: see text] and point out some phenomena concerning this operation.

Resonant-Superlinear Elliptic Problems at High-Order Eigenvalues

This work establishes existence of solution for resonant-superlinear elliptic problems using an appropriate Linking Theorem. The nonlinearity behaves as an asymptotic linear function at $$-\infty $$ (resonant or not) and has a superlinear growth at $$+\infty $$ , with the eventual resonance phenomena occurring in a high order eigenvalue for the associated linear problem. Our main theorems are stated without the well-known Ambrosetti–Rabinowitz condition.

Multiple Solution Results for Perturbed Fractional Differential Equations with Impulses

The multiplicity of classical solutions for impulsive fractional differential equations has been studied by many scholars. Using Morse theory, Brezis and Nirenberg’s Linking Theorem, and Clark theorem, we aim to solve this kind of problems. By this way, we obtain the existence of at least three classical solutions and k distinct pairs of classical solutions. Finally, an example is presented to illustrate the feasibility of the main results in this paper.

Critical Neumann problems with asymmetric nonlinearity

We prove an existence result for a semilinear elliptic equation with superlinear and asymmetric nonlinearity. The asymmetry that we consider is of the type: linear at $-\infty$ and superlinear at $+\infty$. To obtain these results we apply a Linking Theorem.

Subharmonic solutions and minimal periodic solutions of first-order variant subquadratic Hamiltonian systems

Using the homological link theorem and iteration inequalities of Maslov-type index, we prove the multiplicity of subharmonic solutions for some variant subquadratic non-autonomous Hamiltonian systems. Moreover, the minimal period problem has also been considered for the variant subquadratic autonomous Hamiltonian systems.

Existence of weak solutions to superlinear elliptic systems without the Ambrosetti-Rabinowitz condition

In this article, we study the existence of the weak solution for superlinear elliptic equations and systems without the Ambrosetti-Rabinowitz condition. The Ambrosetti-Rabinowitz condition guarantees the boundedness of the PS sequence of the functional I for the corresponding problem. We establish the existence of the weak solution for the superlinear elliptic equation by using \((PS)_c\) form of the Mountain pass lemma, and the existence of the weak solution for the superlinear elliptic system by using \((PS)_c^{*}\) form of the Linking theorem.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/52/abstr.html

A New Proof of the Existence of Nonzero Weak Solutions of Impulsive Fractional Boundary Value Problems

The paper deals with the existence of at least two non zero weak solutions to a new class of impulsive fractional boundary value problems via Brezis and Nirenberg’s Linking Theorem. Finally, an example is presented to illustrate our results.

INFINITELY MANY SOLUTIONS FOR A NONLOCAL PROBLEM

Consider a class of nonlocal problems $ \left\{\begin{array}{lr} -\left(a-b \int_{\Omega}|\nabla u|^{2} d x\right) \Delta u=f(x, u), & x \in \Omega, \\ u=0, & x \in \partial \Omega, \end{array}\right.$ where $ a>0, b>0 $, $ \Omega\subset \mathbb{R}^N $ is a bounded open domain, $ f:\overline{\Omega} \times \mathbb R \longrightarrow \mathbb R $ is a Carath$ \acute{\mbox{e}} $odory function. Under suitable conditions, the equivariant link theorem without the $ (P.S.) $ condition due to Willem is applied to prove that the above problem has infinitely many solutions, whose energy increasingly tends to $ a^2/(4b) $, and they are neither large nor small.

Recovering Fisher-Information from the MGF Alone without Requiring Explicit PMF or PDF from a One-Parameter Exponential Family

It is well-known that a finite moment generating function (m.g.f.) corresponds to a unique probability distribution. So, an important question arises: Is it possible to obtain an expression of Fisher-information, IX(Ɵ); using the m.g.f. alone, that is without requiring explicitly a probability mass function (p.m.f.) or probability density function (p.d.f.), given that the p.m.f. or p.d.f came from a one-parameter exponential family? We revisit the core of statistical inference by developing a clear link (Theorem 1.1) between the m.g.f. and IX(Ɵ). Illustrations are included.

The torsion of real toric manifolds

In view of various results and the nature of their constructions for real toric objects, it is an interesting and also natural problem to know how much torsion can be contained in their cohomology groups. The aim of this paper is to answer this question by explicitly constructing examples of real toric objects to show the richness of torsion which can appear in the cohomology groups with coefficients in a locally constant presheaf. That is, we show that a real quasitoric manifold (or small cover) which plays an important role in the category of real toric objects can have an arbitrary amount of torsion in its cohomology groups with coefficients in a locally constant presheaf. This will be achieved by crucially using the Torsion Theorem for links of Bosio and Meersseman.

Spectral theory approach for a class of radial indefinite variational problems

Considering the radial nonlinear Schrödinger equation(Pr)−Δu+V(x)u=g(x,u) inRN,N≥3 we aim to find a radial nontrivial solution for it, where V changes sign ensuring problem (Pr) is indefinite and g is an asymptotically linear nonlinearity. We work with variational methods associating problem (Pr) to an indefinite functional in order to apply our Abstract Linking Theorem for Cerami sequences in [8] to get a non-trivial critical point for this functional. Our goal is to make use of spectral properties of operator A:=Δ+V(x) restricted to Hrad1(RN), the space of radially symmetric functions in H1(RN), for obtaining a linking geometry structure to the problem and by means of special properties of radially symmetric functions get the necessary compactness.

Existence of three nontrivial solutions for a class of fourth-order elliptic equations

The existence of three nontrivial solutions is established for a class of fourth-order elliptic equations. Our technical approach is based on Linking Theorem and ($\nabla$)-Theorem.

Periodic solutions for critical fractional problems

We deal with the existence of $2\pi$-periodic solutions to the following non-local critical problem \begin{equation*} \left\{\begin{array}{ll} [(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &\mbox{in} (-\pi,\pi)^{N} \\ u(x+2\pi e_{i})=u(x) &\mbox{for all} x \in \mathbb{R}^{N}, \quad i=1, \dots, N, \end{array} \right. \end{equation*} where $s\in (0,1)$, $N \geq 4s$, $m\geq 0$, $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, $W(x)$ is a positive continuous function, and $f(x, u)$ is a superlinear $2\pi$-periodic (in $x$) continuous function with subcritical growth. When $m>0$, the existence of a nonconstant periodic solution is obtained by applying the Linking Theorem, after transforming the above non-local problem into a degenerate elliptic problem in the half-cylinder $(-\pi,\pi)^{N}\times (0, \infty)$, with a nonlinear Neumann boundary condition, through a suitable variant of the extension method in periodic setting. We also consider the case $m=0$ by using a careful procedure of limit. As far as we know, all these results are new.

Critical point theory to isotropic discrete boundary value problems on weighted finite graphs

In this paper, we use the Linking theorem, Clark theorem and Fountain theorem to obtain the existence of solutions to discrete boundary value problem involving the p-Laplacian on simple, connected, undirected and weighted graphs. We obtain three solutions, infinitely many solutions or a number of solutions depending on a parameter and the number of vertices. The eigenvalues of the nonlinear operator are concerned in the study.

Periodic Solutions with Minimal Period for Fourth-Order Nonlinear Difference Equations

A fourth-order nonlinear difference equation is considered. By making use of critical point theory, some new criteria are obtained for the existence of periodic solutions with minimal period. The main methods used are a variational technique and the Linking Theorem.