A nonsmooth Morse-Sard theorem for subanalytic functions

According to the Morse-Sard theorem, any sufficiently smooth function on a Euclidean space remains constant along any arc of critical points. We prove here a theorem of Morse-Sard type suitable as a tool in variational analysis: we broaden the definition of a critical point to the standard notion in nonsmooth optimization, while we restrict the functions under consideration to be semialgebraic or subanalytic. We make no assumption of subdifferential regularity. ?ojasiewicz-type inequalities for nonsmooth functions follow quickly from tools of the kind we develop, leading to convergence theory for subgradient dynamical systems.     

On a global implicit function theorem for locally Lipschitz maps via non-smooth critical point theory

We prove a non-smooth generalization of the global implicit function theorem. More precisely we use the non-smooth local implicit function theorem and the non-smooth critical point theory in order to prove a non-smooth global implicit function theorem for locally Lipschitz functions. A comparison between several global inversion theorems is discussed. Applications to algebraic equations are given.     

Stability theorem and extremum conditions for abnormal problems

We prove a generalized inverse function theorem in a neighborhood of a singular point of a mapping. As corollaries to this theorem, we obtain an inverse function theorem, an error bound theorem, and a tangent cone theorem that extend and strengthen the corresponding classical results in the irregular case. Using these corollaries, we establish necessary extremum conditions that are meaningful for abnormal problems.     

Asymptotic stability and smooth equivalence op ordinary equations : PMM vol.40, n 6, 1976, pp. 1048–1057

Under certain specified conditions the asymptotic stability is a coarse property [1],(i.e. addition of fairly smooth functions to the right-hand sides of equations, does not disturb the asymptotic stability). It is shown below that in this cage the unperturbed system is coarse in a more general sense, namely, any smooth system acted upon by fairly small smooth perturbations, can be returned to its unperturbed state by a smooth reversible transformation. The value and order of the perturbations and the domain of existence of the transformation are all estimated explicitly. The condition required for the above assertion to hold, is that of the existence of a Liapunov function admitting, together with its derivative, specified estimates. This requirement holds, in particular, in the case when the right-hand sides of the unperturbed system are homogeneous functions, the position of equilibrium is asymptotically stable, and its neighborhood contains no solutions bounded when −∞ <t < ∞ (see [1]). If the system is analytic, the requirement will hold in at least all critical cases investigated in which the asymptotic stability with t → ∞ or t → −∞ is fixed, since in these cases the Liapunov function will be analytic, or simply polynomial. It follows therefore from the theorem which we prove, that in all the cases in question, the system is reduced by a smooth transformation, to the polynomial form. If the unperturbed system is linear, then from the theorem proved follows a theorem on linearization appearing in [2]; if the system is nonlinear but of second order, a theorem from [3] ensues. The results obtained in this paper for the nonlinear autonomous systems are extended to the case when the perturbations are continuous and bounded functions of time. This makes possible the investigation of the dynamics of the process in the neighborhood of asymptotically stable equilibria and of periodic modes, ignoring a wide range of external perturbations.     

A Note on the Linearity of Real-valued Functions with Respect to Suitable Metrics

In this paper we prove that for every real-valued Morse function φ on a smooth closed manifold ℳ and every neighborhood U of its critical points a suitable Riemannian metric μ _U exists such that φ is linear outside U     

Circles and Clifford Algebras

Consider a smooth map of a neighborhood of the origin in a real vector space into a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this map takes all lines passing though the origin to the same circles as a Hopf map coming from a representation of a Clifford algebra. We also describe a connection between our result and the Hurwitz–Radon theorem about sums of squares.     

A theorem in fine potential theory and applications to finely holomorphic functions

Consider the map from the fine interior of a compact set to the measures on the fine boundary given by Balayage of the unit point mass onto the fine boundary (the Keldych measure). It is shown that for any point in the domain there is a compact fine neighborhood of the point on which the map is continuous from the initial topology on the compact set to the norm topology on measures. In this paper we only prove a rather special case, the method could easily be generalized to more abstract potential spaces. One consequence of this result is a Hartog-type theorem for finely harmonic functions. We use the Hartog theorem, rational approximation theory, and results proved in a previous paper by the author to prove that the derivative of a finely holomorphic function exists everywhere and is finely holomorphic.     

Global variants of Hartogs’ theorem

Hartogs’ theorem asserts that a separately holomorphic function, defined on an open subset of $$\mathbb {C}^n$$ , is holomorphic in all the variables. We prove a global variant of this theorem for functions defined on an open subset of the product of complex algebraic manifolds. We also obtain global Hartogs-type theorems for complex Nash functions and complex regular functions.     

Local boundary controllability for the semilinear plate equation

weijiu_liu@uow.edu.

The problem of local controllability for the semilinear plate equation with Dirichlet boundary conditions is studied. By making use of Schauder's fixed point theorem and the inverse function theorem, we prove that this system is locally controllable under a super-linear assumption on the nonlinearity, that is, the initial states in a small neighborhood of 0 in a certain function space can be driven to rest by Dirichlet boundary controls. Our super-linear assumption includes the critical exponent.     

A transformational property of the Husimi function and its relation to the wigner function and symplectic tomograms

We consider the Husimi Q-functions, which are quantum quasiprobability distributions in the phase space, and investigate their transformation properties under a scale transformation (q, p) → (λq, λp). We prove a theorem that under this transformation, the Husimi function of a physical state is transformed into a function that is also a Husimi function of some physical state. Therefore, the scale transformation defines a positive map of density operators. We investigate the relation of Husimi functions to Wigner functions and symplectic tomograms and establish how they transform under the scale transformation. As an example, we consider the harmonic oscillator and show how its states transform under the scale transformation.     

Loose saddle points of set-valued maps in topological vector spaces

We give a new existence theorem for loose saddle point of set-valued map having values in a partially ordered topological vector space which is based on continuity and quasiconvexity- quasiconcavity of its scalarized maps. Moreover, we prove a new saddle point theorem for vector-valued functions in locally convex topological vector spaces under weak condition that is the semicontinuity of two function scalarization.     

Multiple Solutions for Nonlinear Coercive Problems with a Nonhomogeneous Differential Operator and a Nonsmooth Potential

We consider a nonlinear elliptic problem driven by a nonlinear nonhomogeneous differential operator and a nonsmooth potential. We prove two multiplicity theorems for problems with coercive energy functional. In both theorems we produce three nontrivial smooth solutions. In the second multiplicity theorem, we provide precise sign information for all three solutions (the first positive, the second negative and the third nodal). Out approach is variational, based on the nonsmooth critical point theory. We also prove an auxiliary result relating smooth and Sobolev local minimizer for a large class of locally Lipschitz functionals.     

Implicit Multifunction Theorems

We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunction theorem by reducing it to an implicit function/solvability theorem for functions. This approach can also be used to prove the Robinson–Ursescu open mapping theorem. As a tool for this alternative proof of the Robinson–Ursescu theorem, we also establish a refined version of the multidirectional mean value inequality which is of independent interest.     

Factorisations of the Helmholtz Operator, Radó’s Theorem, and Clifford Analysis

Radó’s theorem for holomorphic functions asserts that if a continuous function is holomorphic on the complement of its zero locus, then it is holomorphic everywhere. We prove in this paper an equivalent theorem for functions lying in the kernel of a first order differential operator D{\mathcal{D}} such that the Helmholtz operator ∇²+λ can be factorized as the composition [^(D)]D{\widehat{\mathcal{D}}\mathcal{D}} . We also analyse the factorisations [^(D)]D{\widehat{\mathcal{D}}\mathcal{D}} of the Laplace and Helmholtz operators associated to the Clifford analysis and the representations of holomorphic function of several complex variables.     

Proprieétés du degré des morphismes de Belyi

Properties of the degree of Belyi functions. A famous theorem of Belyi characterizes the curves defined over a number field by the existence of an element of its function field with certain ramification properties. In this article we are interested in the degree of these functions. We define the Belyi degree of a curve defined over a number field and the Belyi degree of a point on such a curve. We prove finiteness results concerning these invariants. We give an explicit upper bound for the Belyi degree of a point on the projective line, depending on the height and on the degree of its field of definition.     

Proprieétés du degré des morphismes de Belyi

Properties of the degree of Belyi functions. A famous theorem of Belyi characterizes the curves defined over a number field by the existence of an element of its function field with certain ramification properties. In this article we are interested in the degree of these functions. We define the Belyi degree of a curve defined over a number field and the Belyi degree of a point on such a curve. We prove finiteness results concerning these invariants. We give an explicit upper bound for the Belyi degree of a point on the projective line, depending on the height and on the degree of its field of definition.     

The resolvent of a periodic optical waveguide the continuous spectrum

We consider a two-dimensional optical waveguide with periodic media interface. We study the resolvent of the waveguide in a neighborhood of a purely imaginary point of the spectral parameter. We prove that the resolvent exists on the subspace of functions orthogonal ina certain sense to the singular functions of the continuous spectrum. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 79–89.     

Removal of singularities and analogs of the Sokhotskii–Weierstrass theorem for <Emphasis Type="Italic">Q</Emphasis>-mappings

We prove that an open discrete Q-mapping has a continuous extension to an isolated boundary point if the function Q(x) has finite mean oscillation or logarithmic singularities of order at most n – 1 at this point. Moreover, the extended mapping is open and discrete and is a Q-mapping. As a corollary, we obtain an analog of the well-known Sokhotskii–Weierstrass theorem on Q-mappings. In particular, we prove that an open discrete Q-mapping takes any value infinitely many times in the neighborhood of an essential singularity, except, possibly, for a certain set of capacity zero. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 116–126, January, 2009.     

On integrals constant on congruent domains

We prove that if a real function of two variables is defined, continuous, and bounded on the whole plane, then it is constant under the condition that its integral on each square of unit area is constant. We point out variants of this theorem. We present an example of a function that is not constant but whose integral on each circle of unit radius is constant. Such a function is sin Βx, where Β is any root of the Bessel function J₁. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 183–186, February, 1977.