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In the paper, we consider both finite-dimensional and infinite-dimensional optimization problems with inclusion-type and equality-type constraints. We obtain sufficient conditions for the stability in the weak topology of a solution to this problem with respect to small perturbations of the problem parameters. In the finite-dimensional case, conditions for the stability in the strong topology of the solution are obtained for the problem with equality-type constraints. These conditions are based on a certain implicit function theorem. [ABSTRACT FROM AUTHOR]

We study half-spheres with small radii λ sitting on the boundary of a smooth bounded domain while meeting it orthogonally. Even though it is known that there exist families of CMC and Willmore type half-spheres near a nondegenerate critical point p of the domains boundaries mean curvature, it is unknown in both cases whether these provide a foliation of any deleted neighborhood of p. We prove that this is not guaranteed and establish a criterion in terms of the boundaries geometry that ensures or prevents the respective surfaces from providing such a foliation. This perhaps surprising phenomenon of conditional foliations is absent in the closely related Riemannian setting, where a foliation is guaranteed. We show how this unconditional foliation arises from symmetry considerations and how these fail to apply to the "domain-setting". [ABSTRACT FROM AUTHOR]

This paper considers a mathematical model of steady flows of an inviscid and incompressible fluid moving in the azimuthal direction. The water density varies with depth and the waves are propagating under the force of gravity, over a flat bed and with a free surface, on which acts a force of surface tension. Our solution pertains to large scale equatorial dynamics of a fluid with free surface expressed in cylindrical coordinates. We also prove a regularity result for the free surface. [ABSTRACT FROM AUTHOR]

We consider the area-preserving Willmore evolution of surfaces ϕ that are close to a half-sphere with a small radius, sliding on the boundary S of a domain Ω while meeting it orthogonally. We prove that the flow exists for all times and keeps a "half-spherical" shape. Additionally, we investigate the asymptotic behavior of the flow and prove that for large times the barycenter of the surfaces approximately follows an explicit ordinary differential equation. Imposing additional conditions on the mean curvature of S, we then establish convergence of the flow. [ABSTRACT FROM AUTHOR]

For 1 ≤ p < ∞ , we prove that the dense subspace Y p of ℓ p (Γ) comprising all elements y such that y ∈ ℓ q (Γ) for some q ∈ (0 , p) admits a C ∞ -smooth norm which locally depends on finitely many coordinates. Moreover, such a norm can be chosen as to approximate the · p -norm. This provides examples of dense subspaces of ℓ p (Γ) with a smooth norm which have the maximal possible linear dimension and are not obtained as the linear span of a biorthogonal system. Moreover, when p > 1 or Γ is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in ℓ 1 (Γ) admits a C 1 -smooth norm. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of singular variable-order fractional Kirchhoff problem of the form: where is a bounded domain, is the variable-order fractional Laplacian operator, [u]s(·) is the Gagliardo seminorm and is a continuous and symmetric function. We assume that λ is a non-negative parameter, with and. We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem. [ABSTRACT FROM AUTHOR]

This paper studies well-posedness and parameter sensitivity of the square root LASSO (SR-LASSO), an optimization model for recovering sparse solutions to linear inverse problems in finite dimension. An advantage of the SR-LASSO (e.g., over the standard LASSO) is that the optimal tuning of the regularization parameter is robust with respect to measurement noise. This paper provides three point-based regularity conditions at a solution of the SR-LASSO: the weak, intermediate, and strong assumptions. It is shown that the weak assumption implies uniqueness of the solution in question. The intermediate assumption yields a directionally differentiable and locally Lipschitz solution map (with explicit Lipschitz bounds), whereas the strong assumption gives continuous differentiability of said map around the point in question. Our analysis leads to new theoretical insights on the comparison between SR-LASSO and LASSO from the viewpoint of tuning parameter sensitivity: noise-robust optimal parameter choice for SR-LASSO comes at the "price" of elevated tuning parameter sensitivity. Numerical results support and showcase the theoretical findings. [ABSTRACT FROM AUTHOR]

A celebrated theorem of Serrin asserts that one overdetermined condition on the boundary is enough to obtain radial symmetry in the so-called one-phase overdetermined torsion problem. It is also known that imposing just one overdetermined condition on the boundary is not enough to obtain radial symmetry in the corresponding multi-phase overdetermined problem. In this paper we show that, in order to obtain radial symmetry in the two-phase overdetermined torsion problem, two overdetermined conditions are needed. Moreover, it is noteworthy that this pattern does not extend to multi-phase problems with three or more layers, for which we show the existence of nonradial configurations satisfying countably infinitely many overdetermined conditions on the outer boundary. [ABSTRACT FROM AUTHOR]

The paper extends the empirical likelihood (EL) approach of Liu et al. to a new and very flexible family of latent class models for capture-recapture data also allowing for serial dependence on previous capture history, conditionally on latent type and covariates. The EL approach allows to estimate the overall population size directly rather than by adding estimates conditional to covariate configurations. A Fisher-scoring algorithm for maximum likelihood estimation is proposed and a more efficient alternative to the traditional EL approach for estimating the non-parametric component is introduced; this allows us to show that the mapping between the non-parametric distribution of the covariates and the probabilities of being never captured is one-to-one and strictly increasing. Asymptotic results are outlined, and a procedure for constructing profile likelihood confidence intervals for the population size is presented. Two examples based on real data are used to illustrate the proposed approach and a simulation study indicates that, when estimating the overall undercount, the method proposed here is substantially more efficient than the one based on conditional maximum likelihood estimation, especially when the sample size is not sufficiently large. [ABSTRACT FROM AUTHOR]

For a bounded and smooth enough domain Ω in R n, with n ≥ 2, we consider the problem −∆u = au−β +λh(.,u) in Ω, u = 0 on ∂Ω, u > 0 in Ω, where λ > 0, 0 < β < 3, a ∈ L∞ (Ω), essin f (a) > 0, and with h = h(x,s) ∈C Ω×[0,∞)) positive on Ω×(0,∞) and such that, for any x ∈ Ω, h(x,.) is strictly convex on (0,∞), nondecreasing, belongs to C 2 (0,∞), and satisfies, for some p ∈ (1,n+2/n-2), that lims→∞ hs(x,s) /sp=0 and lims→∞ hs(x,s) /sp=k(x), in both limits uniformly respect to x ∈ Ω, and with k ∈ C (Ω) such that minΩ k > 0. Under these assumptions it is known the existence of Σ >0 such that for λ = 0 and λ = Σ the above problem has exactly a weak solution, whereas for λ ∈ (0, Σ) it has at least two weak solutions, and no weak solutions exist if λ > Σ. For such a Σ we prove that for λ ∈ (0,Σ) the considered problem has it has exactly two weak solutions. [ABSTRACT FROM AUTHOR]

In 1976 Simmons conjectured that every coloring of a 2-dimensional sphere of radius strictly greater than 1/2 in three colors has a pair of monochromatic points at distance 1 apart. We prove this conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence solutions to nonlinear Fučík problems of the form (1) A x = α x + − β x − + g (x) , where A is a symmetric n × n matrix, α , β are real numbers, and g : R n → R n is continuous. The nonlinear problem (1) is motivated by application to nonlinear oscillating systems such as the Tacoma Narrows Bridge. The paper begins by developing a qualitative picture of Fučík spectrum associated with the matrix equation A x = α x + − β x −. In this setting, we present two characterizations: first, we show that under appropriate assumptions the Fučík spectrum consists of curves bifurcating from points (λ , λ) ∈ R 2 , where λ is an eigenvalue of A ; second, we give more global variational characterization of the Fučík curves. In both cases, we present various qualitative properties of the Fučík curves. The paper finishes by presenting two existence theorems for the nonlinear Fučík problem under mild assumptions on the nonlinear term g. [ABSTRACT FROM AUTHOR]

In this article, we present explicit estimates of size of the domain on which the implicit function theorem and the inverse function theorem are valid. For maps that are twice continuously differentiable, these estimates depend upon the magnitude of the first-order derivatives evaluated at the point of interest, and a bound on the second-order derivatives over a region of interest. One of the key contributions of this article is that the estimates presented require minimal numerical computation. In particular, these estimates are arrived at without any intermediate optimization procedures. We then present three applications in optimization and systems and control theory where the computation of such bounds turns out to be important. First, in electrical networks, the power flow operations can be written as quadratically constrained quadratic programs, and we utilize our bounds to compute the size of permissible power variations to ensure stable operations of the power system network. Second, the robustness margin of positive-definite solutions to the algebraic Riccati equation (frequently encountered in control problems) subject to perturbations in the system matrices is computed with the aid of our bounds. Finally, we employ these bounds to provide quantitative estimates of the size of the domains for feedback linearization of discrete-time control systems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of variational free boundary problem. Namely, we say that a free boundary problem is variational if its solutions can be characterized as the critical points of some shape functional. Moreover, we extend the notion of nondegeneracy of a critical point to this setting. As a result, we provide a unified functional-analytical framework that allows us to construct families of solutions to variational free boundary problems whenever the shape functional is nondegenerate at some given solution. As a clarifying example, we apply this machinery to construct families of nontrivial solutions to the two-phase Serrin's overdetermined problem in both the degenerate and nondegenerate case. [ABSTRACT FROM AUTHOR]

In this paper, we study the symmetry properties of nondegenerate critical points of shape functionals using the implicit function theorem. We show that, if a shape functional is invariant with respect to some one-parameter group of rotations, then its nondegenerate critical points (bounded open sets with smooth enough boundary) share the same symmetries. We also consider the case where the shape functional exhibits translational invariance in addition to just rotational invariance. Finally, we study the applications of this result to the theory of one/two-phase overdetermined problems of Serrin-type. En passant, we give a simple proof of the fact that, under suitable smoothness assumptions, the ball is the only nondegenerate critical point of the Lagrangian associated to the maximization problem for the torsional rigidity under a volume constraint. We remark that the proof does not rely on either the method of moving planes or rearrangement techniques. [ABSTRACT FROM AUTHOR]

We study an overdetermined problem that arises as the Euler-Lagrange equation of a weighted variational problem in elasticity. Based on a detailed linear analysis by spherical harmonics, we prove the existence and local uniqueness as well as an optimal stability estimate for the shape of a domain allowing the solvability of the overdetermined problem. Our linear analysis reveals that the solution structure is strongly related to the choice of parameters in the problem. In particular, the global uniqueness holds for the pair of the parameters lying in a triangular region.

We consider families u = u ε , 0 of boundary layer solutions to singularly perturbed quasilinear problems of the type ε 2 (a (x , u (x) , ε) u ′ (x)) ′ = b (x , u (x) , ε) for x ∈ (− 1 , 1) , u (− 1) = u ′ (1) = 0 , and we describe the behaviour of these solution families under small regular, but nonsmooth perturbations, i.e. we show existence and local uniqueness of solutions u = u ε , δ ≈ u ε , 0 to − ε 2 (a (x , u (x) , ε) u ′ (x)) ′ + b (x , u (x) , ε) = δ g (x) for x ∈ (− 1 , 1) , u (− 1) = u ′ (1) = 0 with ε ≈ 0 and δ ≈ 0. Roughly speaking, we show the following: If g is a Dirac function, then for all small ε > 0 and δ ≥ 0 , such that δ / ε is small, those solutions exist, and ‖ u ε , δ − u ε , 0 ‖ ∞ = O (δ / ε) for δ → 0 uniformly with respect to ε. If g ∈ L 2 (− 1 , 1) , then for all small ε > 0 and δ ≥ 0 , such that δ / ε is small, those solutions exist, and ‖ u ε , δ − u ε , 0 ‖ ∞ = O (δ / ε) for δ → 0 uniformly with respect to ε. And if g ∈ L ∞ (− 1 , 1) , then for all small ε > 0 and δ ≥ 0 those solutions exist, and ‖ u ε , δ − u ε , 0 ‖ ∞ = O (δ) for δ → 0 uniformly with respect to ε. Finally we show that these asymptotic estimates are optimal. • We describe the behavior of families of boundary layer solutions to singularly perturbed problems under the influence of small regular perturbations. • The behavior is as follows: The lower the smoothness (with respect to the space variable x) of the perturbation function, the worse the rate (with respect to the perturbation parameters) of convergence of the perturbed boundary layer solution family to the unperturbed one. • We present an abstract result of implicit function theorem type, which is designed for applications to singularly perturbed ODEs and PDEs. [ABSTRACT FROM AUTHOR]

This manuscript investigates a dynamical system in which $ 2N $ primary particles of equal masses move in space under Newton's law of gravitation forming the vertices of antiprisms while a particle of negligible mass moves along the common axis of symmetry of the antiprisms. This $ n $-body problem that we call the restricted hip-hop $ (2N + 1) $-body problem is an extension of the generalized Sitnikov problem studied in [17] for which the primaries remain in a plane. This work also relies on an early study [14] where certain families of periodic hip-hop solutions to a $ 2N $–body problem were constructed. We prove the existence of a continuous symmetric family of solutions of the restricted hip-hop $ (2N+1) $-body problem for each family of symmetric and periodic hip-hop solutions of the primaries studied in [14]. The main tools for proving our results are the implicit function theorem and a compactness argument. In addition, we present some numerical periodic solutions to the restricted $ 7 $-body problem. [ABSTRACT FROM AUTHOR]

This paper formalizes the concepts of weakly and weakly regularly persistent input trajectory as well as their link to the observability Grammian and the existence and uniqueness of solutions of moving horizon estimation (MHE) problems. Additionally, thanks to a new timeuniform implicit function theorem, these notions are proved to imply the stability of MHE solutions with respect to small additive perturbation in the measurements and in the dynamics, both uniformly and nonuniformly in time. Finally, examples and counterexamples of weakly persistent and weakly regularly persistent input trajectories are given in the case of 2D bearing-only navigation. [ABSTRACT FROM AUTHOR]

In this paper, we provide a rigorous computer-assisted proof (CAP) of the conjecture that in the planar four-body problem there exists a unique convex central configuration for any four fixed positive masses in a given order belonging to a closed domain in the mass space. The proof employs the Krawczyk operator and the implicit function theorem (IFT). Notably, we demonstrate that the implicit function theorem can be combined with interval analysis, enabling us to estimate the size of the region where the implicit function exists and extend our findings from one mass point to its neighborhood. [ABSTRACT FROM AUTHOR]

The objective in this work is to propose a novel approach for solving inverse problems from the output space to the input space using automatic differentiation coupled with the implicit function theorem and a path integration scheme. A common way of solving inverse problems in process systems engineering (PSE) and in science, technology, engineering and mathematics (STEM) in general is using nonlinear programming (NLP) tools, which may become computationally expensive when both the underlying process model complexity and dimensionality increase. The proposed approach takes advantage of recent advances in robust automatic differentiation packages to calculate the input space region by integration of governing differential equations of a given process. Such calculations are performed based on an initial starting point from the output space and are capable of maintaining accuracy and reducing computational time when compared to using NLP‐based approaches to obtain the inverse mapping. Two nonlinear case studies, namely a continuous stirred tank reactor (CSTR) and a membrane reactor for conversion of natural gas to value‐added chemicals are addressed using the proposed approach and compared against: (i) extensive (brute‐force) search for forward mapping and (ii) using NLP solvers for obtaining the inverse mapping. The obtained results show that the novel approach is in agreement with the typical approaches, while computational time and complexity are considerably reduced, indicating that a new direction for solving inverse problems is developed in this work. [ABSTRACT FROM AUTHOR]

In a "structured system" of equations, each equation depends on a specified subset of the variables. In this article, we explore properties common to "almost every" system with a fixed structure and how the properties can be read from the corresponding connection graph. A solution p of a system F(p) = c is called robust if it persists despite small changes in F. We establish methods for determining robustness that depends on the structure, as expressed in the properties of the corresponding directed graph of the structured system. The keys to understanding linear and nonlinear structured systems are subsets of variables that we call forward and backward bottlenecks. In particular, when robustness fails in a structured system, it is due to the existence of a unique "backward bottleneck" that we call a "minimax bottleneck." We present a numerical method for locating the minimax bottleneck. We show how to remove it by adding edges to the graph. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to construct one parameter families of embedded, screw motion invariant minimal surfaces in R 3 which limit to parking garage structures. We construct such surfaces by defining Weierstrass data on the quotient and closing the periods. In the nodal limit, the periods reduce to algebraic balance equations for the locations of the helicoidal nodes. For any configuration of nodes that solve the equations and satisfy a nondegeneracy condition, we regenerate to obtain a family of surfaces near the limit. We thus prove the existence of many new examples of surfaces near the nodal limit, with helicoidal or planar ends. Among these are candidates for genus g helicoids distinct from those currently known. We do not require any symmetry for the solutions of the balance equations, which suggests the existence of helicoidal surfaces only symmetric with respect to screw motions. This introduces new directions for the study and classification of screw motion invariant surfaces. [ABSTRACT FROM AUTHOR]

We construct counterexamples to classical calculus facts such as the inverse and implicit function theorems in scale calculus-a generalization of multivariable calculus to infinite-dimensional vector spaces, in which the reparameterization maps relevant to symplectic geometry are smooth. Scale calculus is a corner stone of polyfold theory, which was introduced by Hofer, Wysocki, and Zehnder as a broadly applicable tool for regularizing moduli spaces of pseudoholomorphic curves. We show how the novel nonlinear scale-Fredholm notion in polyfold theory overcomes the lack of implicit function theorems, by formally establishing an often implicitly used fact: The differentials of basic germs-the local models for scale-Fredholm maps-vary continuously in the space of bounded operators when the base point changes. We moreover demonstrate that this continuity holds only in specific coordinates, by constructing an example of a scale-diffeomorphism and scale-Fredholm map with discontinuous differentials. This justifies the high technical complexity in the foundations of polyfold theory.

Differential-algebraic equations (DAEs) are widely used for modelling dynamical systems. In the numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing steps prior to numerical integration. Existing DAE solvers commonly adopt structural preprocessing methods based on combinatorial optimization. Unfortunately, structural methods fail if the DAE has a singular system Jacobian matrix. For such DAEs, methods have been proposed to modify them to other DAEs to which structural methods are applicable, based on the combinatorial relaxation technique. Existing modification methods, however, work only for DAEs that are linear or close to linear. This paper presents two new modification methods for nonlinear DAEs: the substitution method and the augmentation method. Both methods are based on the combinatorial relaxation approach and are applicable to a large class of nonlinear DAEs. The substitution method symbolically solves equations for some derivatives based on the implicit function theorem and substitutes the solution back into the system. Instead of solving equations, the augmentation method modifies DAEs by appending new variables and equations. Our methods are implemented as a MATLAB library using MuPAD, and through its application to practical DAEs, we show that our methods can be used as a promising preprocessing of DAEs that the index reduction procedure in MATLAB cannot handle. [ABSTRACT FROM AUTHOR]

We derive and subsequently analyze an exact solution of the geophysical fluid dynamics equations which describes equatorial flows (in spherical coordinates) and has a discontinuous fluid stratification that varies with both depth and latitude. More precisely, this solution represents a steady, purely–azimuthal equatorial two-layer flow with an associated free-surface and a discontinuous distribution of the density which gives rise to an interface separating the two fluid regions. While the velocity field and the pressure are given by means of explicit formulas, the shape of the free surface and of the interface are given in implicit form: indeed we demonstrate that there is a well-defined relationship between the imposed pressure at the free-surface and the resulting distortion of the surface's shape. Moreover, imposing the continuity of the pressure along the interface generates an equation that describes (implicitly) the shape of the interface. We also provide a regularity result for the interface defining function under certain assumptions on the velocity field. [ABSTRACT FROM AUTHOR]

The maximal entropy OWA operator (MEOWA) weights can be obtained by solving a nonlinear programming problem with a linear constraint for the level of orness. Since the exact MEOWA weights are not known for the general case we can only find approximate solutions. We will prove that the nonlinear programming problem for obtaining MEOWA weights is well-posed: it has a unique solution and each MEOWA weight changes continuously with the initial level of orness. Using the implicit function theorem we will show that MEOWA weights are Lipschitz-continuous functions of the orness level. The stability property of the MEOWA weights under small changes of the orness level guarantees that small rounding errors of digital computation and small errors of measurement of the orness level can cause only a small deviation in MEOWA weights, i.e. every successive approximation method can be applied to the computation of the approximation of the exact MEOWA weights. [ABSTRACT FROM AUTHOR]

We study how a general steady configuration of finitely many point vortices, with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady configuration of vortex patches. The configurations can be uniformly rotating, uniformly translating, or completely stationary. Using a technique first introduced by Hmidi and Mateu [Comm. Math. Phys., 350 (2017), pp. 699-747] for vortex pairs, we reformulate the problem for the patch boundaries so that it no longer appears singular in the point vortex limit. Provided the point vortex equilibrium is nondegenerate in a natural sense, solutions can then be constructed directly using the implicit function theorem, yielding asymptotics for the shape of the patch boundaries. As an application, we construct new families of asymmetric translating and rotating pairs, as well as stationary tripoles. We also show how the techniques can be adapted for highly symmetric configurations such as regular polygons, body-centered polygons, and nested regular polygons by integrating the appropriate symmetries into the formulation of the problem. [ABSTRACT FROM AUTHOR]

Previous Mizar articles [7, 6, 5] formalized the implicit and inverse function theorems for Frechet continuously differentiable maps on Banach spaces. In this paper, using the Mizar system [1], [2], we formalize these theorems on Euclidean spaces by specializing them. We referred to [4], [12], [10], [11] in this formalization. [ABSTRACT FROM AUTHOR]

The inverse eigenvalue problem of a graph studies the real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of the graph. The strong spectral property (SSP) is an important tool for this problem. This note establishes the bifurcation lemma, which states that if a spectrum can be realized by a matrix with the SSP for some graph, then all the nearby spectra can also be realized by matrices with the SSP for the same graph. The idea of the bifurcation lemma also works for other strong properties and for not necessarily symmetric matrices. This is used to develop new techniques for verifying a spectrally arbitrary pattern or inertially arbitrary pattern. The bifurcation lemma provides a unified theoretical foundation for several known results, such as the stable northeast lemma and the nilpotent-centralizer method. [ABSTRACT FROM AUTHOR]

In mathematical modeling, it is common to have an equation F(p)=c, where the exact form of F is not known. This article shows that there are large classes of F where almost all F share the same properties. The classes we investigate are vector spaces F of C¹ functions F:RN→RM that satisfy the following condition: F has "almost constant rank" (ACR) if there is a constant integer ρ(F)≥0 such that rank (DF(p))=ρ(F) for "almost every" F∈F and almost every p∈RN. If the vector space F is finite dimensional, then "almost every" is with respect to the Lebesgue measure on F, and otherwise, it means almost every in the sense of prevalence, as described herein. Most function spaces commonly used for modeling purposes are ACR. In particular, we show that if all of the functions in F are linear or polynomial or real analytic, or if F is the set of all functions in a "structured system", then F is ACR. For each F and p, the solution set of p∈RN is SolSet (p):={x:F(x)=F(p)}. A solution set of F(p)=c is called robust if it persists despite small changes in F and p. The following two global results are proved for almost every F in an ACR vector space F: (1) Either the solution set SolSet (p) is robust for almost every p∈RN, or none of the solution sets are robust. (2) The solution set SolSet (p) is a C∞ -manifold of dimension d=N-ρ(F). In particular, d is the same for almost every F∈F. [ABSTRACT FROM AUTHOR]

In this article, a new efficient numerical method is proposed to solve control theory problems in a finite difference setting. The method was originally devised by Gigena for determining and analyzing constrained local extrema without using Lagrange multipliers with the use of differential forms. He efficiently solved a number of algebraic problems and proved his method to be a better alternative as compared with already existing methods. Inspired by his work, we extended the scope of his work and generalized his technique to solve optimal control problems. The results are highly encouraging, and it can be seen that the proposed method has great advantages in terms of efficiency and computational cost in solving optimal control problems. This new approach provides a new way of looking at solving numerous other optimal control problems, such as discrete time models, with ease. [ABSTRACT FROM AUTHOR]

We prove a result characterizing conditions for the existence and uniqueness of solutions of a certain Diophantine equation, then using techniques from equivariant bifurcation theory, we apply the result to prove symmetric Hopf bifurcation type theorems for both dissipative and non-dissipative autonomous wave equations, for a large set of spatial dimensions. For the latter only the classical implicit function theorem is used. The set of admissible spatial dimensions is the union of the perfect squares together with finitely many non-perfect squares. [ABSTRACT FROM AUTHOR]

We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type ∂ t 2 u (t , x) - a (x , λ) 2 ∂ x 2 u (t , x) = b (x , λ , u (t , x) , u (t - τ , x) , ∂ t u (t , x) , ∂ x u (t , x)) , x ∈ (0 , 1) with smooth coefficient functions a and b such that a (x , λ) > 0 and b (x , λ , 0 , 0 , 0 , 0) = 0 for all x and λ . We state conditions ensuring Hopf bifurcation, i.e., existence, local uniqueness (up to time shifts), regularity (with respect to t and x) and smooth dependence (on τ and λ ) of small non-stationary time-periodic solutions, which bifurcate from the stationary solution u = 0 , and we derive a formula which determines the bifurcation direction with respect to the bifurcation parameter τ . To this end, we transform the wave equation into a system of partial integral equations by means of integration along characteristics and then apply a Lyapunov-Schmidt procedure and a generalized implicit function theorem. The main technical difficulties, which have to be managed, are typical for hyperbolic PDEs (with or without delay): small divisors and the "loss of derivatives" property. We do not use any properties of the corresponding initial-boundary value problem. In particular, our results are true also for negative delays τ . [ABSTRACT FROM AUTHOR]

A two-dimensional inclusion of core–shell structure is neutral to multiple uniform fields if and only if the core and the shell are concentric disks, provided that the conductivity of the matrix is isotropic. An inclusion is said to be neutral if upon its insertion the uniform field is not perturbed at all. In this paper, we consider inclusions of core–shell structure of general shape which are weakly neutral to multiple uniform fields. An inclusion is said to be weakly neutral if the field perturbation is mild. We show, by an implicit function theorem, that if the core is a small perturbation of a disk, then we can coat it by a shell so that the resulting structure becomes weakly neutral to multiple uniform fields. [ABSTRACT FROM AUTHOR]

Adaptive two-stage designs for clinical trials are well understood from a statistical perspective. However, there is still few research on how the stage-two sample size looks like when it is regarded as a function of the first-stage test statistic. In this paper, a formal proof on the concavity of the sample size function is provided if the design's second stage is optimized such that it minimizes the expected sample size under the alternative under constraints on maximal type I error rate and minimal power. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behavior of a mathematical model of cancer including tumor cells, immune cells, and normal cells is investigated when a delay term is induced. Though the model was originally proposed by De Pillis et al. (Math. Comput. Model. 37:1221–1244, 2003), to make the model more realistic, we have added a delay term into the model, and it has incorporated novelty in our present work. The stability of existing equilibrium points in the delay-induced system is studied in detail. Global stability conditions of the tumor-free equilibrium point have been found. It is shown that due to this delay effect, the coexisting equilibrium point may lose its stability through a Hopf bifurcation. The implicit function theorem is applied to characterize a complex function in a neighborhood of delay terms. Additionally, the presence of Hopf bifurcation is demonstrated when the transversality conditions are satisfied. The length of delay for which the solutions preserve the stability of the limit cycle is estimated. Finally, through a series of numerical simulations, the theoretical results are formally examined. [ABSTRACT FROM AUTHOR]

The implicit function theorem (IFT) offers a way of deriving a correspondence between the parameter space and the Nash equilibria of a game. However, which equilibrium will actually emerge after a parameter change involves a dynamic adjustment process, which may significantly differ from IFT predictions. Utilizing the notion of local uniform contraction mappings, we show that IFT predictions are consistent with economic behavior at locally contraction stable equilibria, which is both a necessary and sufficient condition in games of strategic complements. When best response functions are monotone, we can address the convergence of play under more general adaptive dynamics. • Implicit function theorem predicts which equilibrium emerges after parameter shift. • These predictions may differ from results of actual dynamic adjustment processes. • We show that both are consistent at locally contraction stable equilibria. • This is a necessary and sufficient condition in games of strategic complements. • For monotone best responses, we address convergence under more general dynamics. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the egg-sperm chemotaxis model of coral with the incompressible fluid equations in the whole space. The existence of global mild solutions in scaling invariant spaces is proved with sufficient small initial data. Here the main tool we use is the implicit function theorem. Furthermore, we obtain the asymptotic stability of solutions when the time goes to infinity. Since the initial data could be in the weak L p $L^{p}$ -spaces, we finally get the existence of self-similar solutions when the initial data are small homogeneous functions.

The paper illustrates connections between classical results of Analysis and Optimization. The focus is on new elementary proofs of Implicit Function Theorem, Lusternik's Theorem, and optimality conditions for equality constrained optimization problems. The proofs are based on Fermat's Theorem and the Weierstrass Theorem and do not use the contraction mapping principle or other advanced results of Real Analysis, so they can be used in any introductory course on Optimization or Real Analysis without the requirement of the advanced background in analysis. The paper also presents a simple proof of Implicit Function Theorem in normed linear spaces. [ABSTRACT FROM AUTHOR]