Finite-dimensional Perron effect of change of all values of characteristic exponents of differential systems

We obtain a finite-dimensional Perron effect of change of values λ ₁ ≤ … ≤ λ _n < 0 of all arbitrarily specified negative characteristic exponents of the n-dimensional system of linear approximation with infinitely differentiable bounded coefficients to arbitrarily specified, arranged in ascending order, values β _k ≥ λ _k , k = 1, …, n, of characteristic exponents of all nontrivial solutions of an n-dimensional nonlinear differential system with an infinitely differentiable perturbation of arbitrary order m > 1 of smallness in a neighborhood of the origin and growth outside it. Each value β _k is realized by all nontrivial solutions of the perturbed system issuing from the difference R ^k |R ^k?1 of embedded subspaces R ¹ ? R ² ? … ? R ⁿ .     

Continual version of the Perron effect of change of values of the characteristic exponents

We prove the existence of a perturbed two-dimensional system of ordinary differential equations such that its linear approximation has arbitrarily prescribed negative characteristic exponents, the perturbation is of arbitrarily prescribed higher order of smallness in a neighborhood of the origin, all of its nontrivial solutions are infinitely extendible to the right, and the whole set of their Lyapunov exponents is contained in the positive half-line, is bounded, and has positive Lebesgue measure. In the general case, we also obtain explicit representations of the exponents of these solutions via their initial values.     

On the existence of linear Pfaff systems with lower characteristic sets of positive Lebesgue <Emphasis Type="Italic">m</Emphasis>-measure

We prove the existence of an n-dimensional completely integrable Pfaff system with multidimensional time of dimension m ? 2, with bounded infinitely differentiable coefficients, and with the set of lower characteristic vectors of its solutions having positive Lebesgue m-measure.     

On the Baire Classification of Positive Characteristic Exponents in the Perron Effect of Change of Their Values

In the complete Perron effect of change of values of characteristic exponents, where all nontrivial solutions y(t, y₀) of the perturbed two-dimensional differential system are infinitely extendible and have finite positive exponents (the exponents of the linear approximation system being negative), we prove that the Lyapunov exponent λ[y(·, y₀)] of these solutions is a function of the second Baire class of their initial vectors y₀ ∈ ?ⁿ {0}.     

Sign reversal of the characteristic exponents of solutions of a differential system with initial data on finitely many points and lines

We realize a version of the Perron sign reversal effect for the characteristic exponents of a two-dimensional differential system; the exponents are negative for the linear approximation system and positive for the nontrivial solutions of the full nonlinear system with a higher-order perturbation in a neighborhood of the origin and with initial data on an arbitrary finite set of points and lines on the plane R ².     

Einstein structures: Existence versus uniqueness

Two well-known questions in differential geometry are “Does every compact manifold of dimension greater than four admit an Einstein metric?” and “Does an Einstein metric of a negative scalar curvature exist on a sphere?” We demonstrate that these questions are related: For everyn≥5 the existence of metrics for which the deviation from being Einstein is arbitrarily small on every compact manifold of dimensionn (or even on every smooth homology sphere of dimensionn) implies the existence of metrics of negative Ricci curvature on the sphereS ⁿ for which the deviation from being Einstein is arbitrarily small. Furthermore, assuming either a version of the Palais-Smale condition or the plausible looking existence of an algorithm deciding when a given metric on a compact manifold is close to an Einstein metric, we show for anyn≥5 that: 1) If everyn-dimensional smooth homology sphere admits an Einstein metric thenS ⁿ admits infinitely many Einstein structures of volume one and of negative scalar curvature; 2) If every compactn-dimensional manifold admits an Einstein metric then every compactn-dimensional manifold admits infinitely many distinct Einstein structures of volume one and of negative scalar curvature.     

On the perron sign change effect for Lyapunov characteristic exponents of solutions of differential systems

The Perron effect is the effect in which the characteristic Lyapunov exponents of solutions of a differential system change sign from negative to positive when passing to a perturbed system. We show that this effect is realized on all nontrivial solutions of two two-dimensional systems: an original linear system with negative characteristic exponents and a perturbed system with small perturbations of arbitrary order m > 1 in a neighborhood of the origin, all of whose nontrivial solutions have positive characteristic exponents. We compute the exact positive value of the characteristic exponents of solutions of the two-dimensional nonlinear Perron system with small second-order perturbations, which realizes only a partial Perron effect.     

Construction of a linear Pfaff system with <Emphasis Type="Italic">m</Emphasis>-dimensional time and with given characteristic set and lower characteristic set

For any positive integers n ≥ 1 and m ≥ 2, we give a constructive proof of the existence of linear n-dimensional Pfaff systems with m-dimensional time and with infinitely differentiable coefficient matrices such that the characteristic and lower characteristic sets of these systems are given sets that are the graphs of a concave continuous function and a convex continuous function, respectively, defined and monotone decreasing on simply connected closed bounded convex domains of the space ?^m?1.     

Generalization of the perron effect whereby the characteristic exponents of all solutions of two differential systems change their sign from negative to positive

We obtain a generalization of the complete Perron effect whereby the characteristic exponents of all solutions change their sign from negative for the linear approximation system to positive for a nonlinear system with perturbations of higher-order smallness [Differ. Uravn., 2010, vol. 46, no. 10, pp. 1388–1402]. Namely, for arbitrary parameters λ ₁ ≤ λ ₂ < 0 and m > 1 and for arbitrary intervals [b _i , d _i ) ⊂ [λ _i ,+∞), i = 1, 2, with boundaries d ₁ ≤ b ₂, we prove the existence of (i) a two-dimensional linear differential system with bounded coefficient matrix A(t) infinitely differentiable on the half-line t ≥ 1 and with characteristic exponents λ ₁(A) = λ ₁ ≤ λ ₂(A) = λ ₂ < 0; (ii) a perturbation f(t, y) of smallness order m > 1 infinitely differentiable with respect to time t > 1 and continuously differentiable with respect to y ₁ and y ₂, y = (y ₁, y ₂) ∈ R ² such that all nontrivial solutions y(t, c), c ∈ R ², of the nonlinear system .y = A(t)y + f(t, y), y ∈ R ², t ≥ 1, are infinitely extendible to the right and have characteristic exponents λ[y] ∈ [b ₁, d ₁) for c ₂ = 0 and λ[y] ∈ [b ₂, d ₂) for c ₂ ≠ 0.     

On positive solutions of perturbed nonlinear differential equations

Some results are given concerning positive solutions of equations of the form x⁽ⁿ⁾ + P(t) G(x) = Q(t, x).Let class I (II) consist of all n-times differentiable functions x(t), such that x(t)>0 and x^{(n ? 1)}(t) ? 0 (x^{(n ? 1)}(t) ? 0) for all large t. Two theorems are given guaranteeing the nonexistence of solutions in class I and II, respectively, and three theorems ensure the convergence to zero of positive solutions. A recent result of Hammett concerning the second-order case is extended to the general case.     

On basicity of perturbed system of exponents in Lebesgue space with variable summability factor

In this paper, the system of exponents \(\{ e^{i\lambda _n t} \} _{n \in Z} \) is considered, where {λ _n } ? R has the following asymptotic form: λ _n = n ? α sign n + O(|n|^?β). Basis properties of this system in Lebesgue space with variable summability factor are investigated.     

The existence of k-convex hypersurface with prescribed mean curvature

Using the strong maximum principle, we obtain a constant rank theorem for the k-convex solutions of semilinear elliptic partial differential equations. As an application we obtain an existence theorem of k-convex starshaped hypersurface with prescribed mean curvature in R ⁿ⁺¹.     

Existence of linear Pfaffian systems whose lower characteristic set has positive measure in <Emphasis Type="Italic">R</Emphasis><Superscript>3</Superscript>

We prove the existence of a completely integrable Pfaffian system ?x/?t i = A _i (t)x, x ∈ R ⁿ , t = (t ₁, t ₂, t ₃) ∈ R ₊ ³ , i = 1, 2, 3, with infinitely differentiable bounded coefficients and with lower characteristic set of positive three-dimensional Lebesgue measure.     

Stable Periodic Solutions of Periodic Systems of Differential Equations

An infinitely differentiable periodic two-dimensional system of differential equations is considered. It is assumed that there is a hyperbolic periodic solution and there exists a homoclinic solution to the periodic solution. It is shown that, for a certain type of tangency of the stable manifold and unstable manifold, any neighborhood of the nontransversal homoclinic solution contains a countable set of stable periodic solutions such that their characteristic exponents are separated from zero.     

Finitely generated submodules of differentiable functions II

Let [$\text{E}$(Ω)]^p be the Cartesian product of the space of real-valued infinitely differentiable functions on a connected open set Ω in $\text{R}$ⁿ with itself p-times. The finitely generated submodules of [$\text{E}$(Ω)]^p are of the form im(F) where F: [$\text{E}$(Ω)]^q → [$\text{E}$(Ω)]^p is a p × q matrix of infinitely differentiable functions on Ω. Let $\text{r =}\text{max}\text{{}\text{rank}\text{(F(x)): x ? Ω}}$. The main results of the present paper are that for Ω ? $\text{R}$ⁿ, if the finitely generated submodule im(F) is closed in [$\text{E}$(Ω)]^p, then for every x?ω with rank(F(x)) < r there exists an r × r sub-matrix A of F such that x is a zero of finite order of det(A), and for Ω ? $\text{R}$¹ the converse also holds.     

Realization of the Perron effect whereby the characteristic exponents of solutions of differential systems change their values

We realize the Perron effect of change of values of characteristic exponents: for arbitrary parameters λ ₁ <- λ ₂ < 0, β ₂ ≥ β ₁ ≥ λ ₂, and m > 1, we prove the existence of a linear differential system $ \dot x $ \dot x = A(t)x, x ∈ R ₂, t ≥ t ₀, with bounded infinitely differentiable coefficients and with characteristic exponents λ ₁(A) = λ ₁ <- λ ₂(A) = λ ₂ and of an m-perturbation f: [t ₀,+∞) × R ₂ → R ₂ infinitely differentiable in time, continuously differentiable with respect to the phase variables y ₁ and y ₂, (y ₁, y ₂) = y ∈ R ₂ (infinitely differentiable with respect to the variables y ₁ ≠ 0 and y ₂ ≠ 0 and with respect to all of these variables in the case of a positive integer m > 1), satisfying the condition ‖f(t, y)‖ ≤ const × ‖y‖ ^m , y ∈ R ², t ≥ t ₀, and such that all nontrivial solutions y(t, c) of the perturbed system

Operator-stable distributions and stable marginals

Sharpe has shown that full operator-stable distributions μ on Rⁿ are infinitely divisible and for a suitable automorphism B depending on μ satisfy the relation $\text{μ}{}^{\mathrm{t}}\text{= μt}{}^{\mathrm{?B}}\text{1 δ(b(t))}$ for all t > 0. B is called an exponent for μ. It is proved here that if an operator-stable distribution on Rⁿ has n linearly independent univariate stable marginals, then its exponents are semi-simple operators. In addition necessary and sufficient conditions are given for such a distribution on R² to have univariate stable marginals. The proofs use a hitherto unpublished result of Sharpe's that all full operator-stable distributions are absolutely continuous. His proof is provided here.     

Edge-Ramsey theory

LetK be a configuration, a set of points in some finite-dimensional Euclidean space. Letn andk be positive integers. The notationR(K, n, r) is an abbreviation for the following statement: For everyr-coloring of the points of then-dimensional Euclidean space,R ⁿ , a monochromatic configurationL which is congruent toK exists. A configurationK is Ramsey if the following holds: For every positive integerr, a positive integern=n(K, r) exists such that, for allm≥n, R(K, m, r) holds. A configuration is spherical if it can be embedded in the surface of a sphere inn-space, providedn is sufficiently large. It is relatively easy to show that if a configuration is Ramsey, it must be spherical. Accordingly, a good fraction of the research efforts in Euclidean Ramsey theory is devoted to determining which spherical configurations are Ramsey. It is known that then-dimensional measure polytopes (the higher-dimensional analogs of a cube), then-dimensional simplex, and the regular polyhedra inR ² andR ³ are Ramsey. Now letE denote a set of edges in a configurationK. The pair (K, E) is called an edge-configuration, andR _e (K, E, n, r) is used as an abbreviation for the following statement: For anyr-coloring of the edges ofR ⁿ , there is an edge configuration (L, F) congruent to (K, E) so that all edges inF are assigned the same color. An edge-configuration isedge-Ramsey if, for allr≥1, a positive integern=n(K, E, r) exists so that ifm≥n, the statementR _e (K, E, m, r) holds. IfK is a regular polytope, it is saidK isedge-Ramsey when the configuration determined by the set of edges of minimum length is edge-Ramsey. It is known that then-dimensional simplex is edge-Ramsey and that the nodes of any edge-Ramsey configuration can be partitioned into two spherical sets. Furthermore, the edges of any edge-Ramsey configuration must all have the same length. It is conjectured that the unit square is edge-Ramsey, and it is natural to ask the more general question: Which regular polytopes are edge-Ramsey? In this article it is shown that then-dimensional measure polytope and then-dimensional cross polytope are edge-Ramsey. It is also shown that these two infinite families and then-dimensional simplexes are the only regular edge-Ramsey polytopes, with the possible exceptions of the hexagon and the 24-cell.     

Non-oscillating Paley-Wiener functions

A non-oscillating Paley-Wiener function is a real entire functionf of exponential type belonging toL ₂(R) and such that each derivativef ⁽ⁿ⁾,n=0, 1, 2,…, has only a finite number of real zeros. It is established that the class of such functions is non-empty and contains functions of arbitrarily fast decay onR allowed by the convergence of the logarithmic integral. It is shown that the Fourier transform of a non-oscillating Paley-Wiener function must be infinitely differentiable outside the origin. We also give close to best possible asymptotic (asn→∞) estimates of the number of real zeros of then-th derivative of a functionf of the class and the size of the smallest interval containing these zeros.