Generalized varieties of sums of powers

Let X ? P^N be an irreducible, non-degenerate variety. The generalized variety of sums of powers V S P_H^X(h) of X is the closure in the Hilbert scheme Hilb_h (X) of the locus parametrizing collections of points {x₁,..., x_h} such that the (h -1)-plane >x₁,..., x_h> passes through a fixed general point p ∈ P^N. When X = V_dⁿ is a Veronese variety we recover the classical variety of sums of powers V S P(F, h) parametrizing additive decompositions of a homogeneous polynomial as powers of linear forms. In this paper we study the birational behavior of V S P_H^X(h). In particular, we show how some birational properties, such as rationality, unirationalityand rational connectedness, of V S P_H^X(h) are inherited from the birational geometry of variety X itself.     

On the Surface Integral Approximation of the Second Derivatives of the Potential of a Bulk Charge Located in a Layer of Small Thickness

We consider a bulk charge potential of the form

$$u(x) = \int\limits_\Omega {g(y)F(x - y)dy,x = ({x_1},{x_2},{x_3}) \in {\mathbb{R}^3},} $$

where Ω is a layer of small thickness h > 0 located around the midsurface Σ, which can be either closed or open, and F(x ? y) is a function with a singularity of the form 1/|x ? y|. We prove that, under certain assumptions on the shape of the surface Σ, the kernel F, and the function g at each point x lying on the midsurface Σ (but not on its boundary), the second derivatives of the function u can be represented as

$$\frac{{{\partial ^2}u(x)}}{{\partial {x_i}\partial {x_j}}} = h\int\limits_\Sigma {g(y)\frac{{{\partial ^2}F(x - y)}}{{\partial {x_i}\partial {x_j}}}} dy - {n_i}(x){n_j}(x)g(x) + {\gamma _{ij}}(x),i,j = 1,2,3,$$

where the function γ_ij(x) does not exceed in absolute value a certain quantity of the order of h², the surface integral is understood in the sense of Hadamard finite value, and the n_i(x), i = 1, 2, 3, are the coordinates of the normal vector on the surface Σ at a point x.

     

Semiclassical resonances associated with a periodic orbit

We consider resonances for a h-pseudo-differential operator H(x, hD _x; h) induced by a periodic orbit of hyperbolic type. We generalize the framework of Gérard and Sjöstrand, in the sense that we allow hyperbolic and elliptic eigenvalues of the Poincarémap, and look for so-called semi-excited resonances with imaginary part of magnitude ?h log h, or h ^δ, with 0 < δ < 1.     

On stabilizability of evolution systems of partial differential equations on ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×[0,+∞) by time-delayed feedback controlsby time-delayed feedback controls

In this work we obtain sufficient conditions for stabilizability by time-delayed feedback controls for the system

$\frac{{\partial w\left( {x,t} \right)}}{{\partial t}} = A(D_x )w(x,t) - A(D_x )u(x,t), x \in \mathbb{R}^n , t > h, $

where D _x =(-i?/?x ₁,...-i?/?x _n ), A(σ) and B(σ) are polynomial matrices (m×m), det B(σ)≡0 on ? ⁿ , w is an unknown function, u(·,t)=P(D _x )w(·,t?h) is a control, h>0. Here P is an infinite differentiable matrix (m×m), and the norm of each of its derivatives does not exceed Γ(1+|σ|²)^γ for some Γ, γ∈? depending on the order of this derivative. Necessary conditions for stabilizability of this system are also obtained. In particular, we study the stabilizability problem for the systems corresponding to the telegraph equation, the wave equation, the heat equation, the Schrödinger equation and another model equation. To obtain these results we use the Fourier transform method, the Lojasiewicz inequality and the Tarski—Seidenberg theorem and its corollaries. To choose an appropriate P and stabilize this system, we also prove some estimates of the real parts of the zeros of the quasipolynomial det {Iλ-A(σ)+B(σ)P(σ)e ^-hλ.

     

Hamiltonian cycles in Dirac graphs

We prove that for any n-vertex Dirac graph (graph with minimum degree at least n/2) G=(V,E), the number, Ψ(G), of Hamiltonian cycles in G is at least

$exp_2 [2h(G) - n\log e - o(n)],$

where h(G)=maxΣ _e x _e log(1/x _e ), the maximum over x: E → ?⁺ satisfying Σ _e?υ x _e = 1 for each υ ∈ V, and log =log₂. (A second paper will show that this bound is tight up to the o(n).)

We also show that for any (Dirac) G of minimum degree at least d, h(G) ≥ (n/2) logd, so that Ψ(G) > (d/(e + o(1))) ⁿ . In particular, this says that for any Dirac G we have Ψ(G) > n!/(2 + o(1)) ⁿ , confirming a conjecture of G. Sárközy, Selkow, and Szemerédi which was the original motivation for this work.     

One Approach to the Computation of Asymptotics of Integrals of Rapidly Varying Functions

We consider integrals of the form

$$I\left( {x,h} \right) = \frac{1}{{{{\left( {2\pi h} \right)}^{k/2}}}}\int_{{\mathbb{R}^k}} {f\left( {\frac{{S\left( {x,\theta } \right)}}{h},x,\theta } \right)} d\theta $$

, where h is a small positive parameter and S(x, θ) and f(τ, x, θ) are smooth functions of variables τ ∈ ?, x ∈ ? ⁿ , and θ ∈ ? ^k ; moreover, S(x, θ) is real-valued and f(τ, x, θ) rapidly decays as |τ| →∞. We suggest an approach to the computation of the asymptotics of such integrals as h → 0 with the use of the abstract stationary phase method.

     

On the length of a finite group and of its 2-generator subgroups

The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series of G each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group G is the least number h = h* (G) such that F* _h (G) = G, where F* ₁ (G) = F* (G) is the generalized Fitting subgroup, and F* _i+1(G) is the inverse image of F* (G/F*_i (G)). In the present paper we prove that if λ(J) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (〈x, x^g 〉) ≤ k for allx, g ∈ G such that 〈x, x^g 〉 is soluble, then h* (G) is k-bounded.     

On boundary value problems for systems of nonlinear generalized ordinary differential equations

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R ⁿ → R ⁿ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → R ^n×n with bounded total variation components, and h: BV_s([a, b],R ⁿ ) → R ⁿ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t₁(x)) = B(x) · x(t ₂(x))+c ₀, where t _i: BV_s([a, b],R ⁿ ) → [a, b] (i = 1, 2) and B: BV_s([a, b], R ⁿ ) → R ⁿ are continuous operators, and c ₀ ∈ R ⁿ .     

On a direct method for proving the Hyers-Ulam stability of functional equations

We study the stability of an equation in a single variable of the form

f(x)=af(h(x))+bf(−h(x))     

Solvability of a Kind of Sturm–Liouville Boundary Value Problems with Impulses via Variational Methods

The purpose of this paper is to use a three critical point theorem due to Ricceri to obtain the existence of at least three solutions for the following Sturm–Liouville boundary value problem with impulses

$\begin{cases}(\phi_{p}(x'(t)))'=(a(t)\phi_{p}(x)+\lambda f(t,x)+\mu h(x))g(x'(t)),\quad \mbox{a.e. }t\in[0,1],\\\Delta G(x'(t_{i}))=I_{i}(x(t_{i})),\quad i=1,2,\ldots,k,\\\alpha_{1}x(0)-\alpha_{2}x'(0)=0,\\\beta_{1}x(1)+\beta_{2}x'(1)=0,\end{cases}$

where p>1, φ _p (x)=|x|^p?2 x, λ, μ are positive parameters, \(G(x)=\int_{0}^{x}\frac{(p-1)|s|^{p-2}}{g(s)}\,ds\). The interest is that the nonlinear term includes x′. We exhibit the existence of at least three solutions and h(x) can be an arbitrary C ¹ functional with compact derivative. As an application, an example is given to illustrate the results.

     

Scattering for the quartic generalised Korteweg-de Vries equation

We show that the quartic generalised KdV equation

ut+uxxx+(u⁴x₎=0     

Global existence of real roots and random Newton flow algorithm for nonlinear system of equations

To solve nonlinear system of equation, F(x) = 0, a continuous Newton flow x _t (t) = V (x) = ?(DF(x))^?1 F(x), x(0) = x ⁰ and its mathematical properties, such as the central field, global existence and uniqueness of real roots and the structure of the singular surface, are studied. We concisely introduce random Newton flow algorithm (NFA) for finding all roots, based on discrete Newton flow x ^j+1 = x ^j + hV (x ^j ) with random initial value x ⁰ and h ∈ (0, 1], and three computable quantities, g _j , d _j and K _j . The numerical experiments with dimension n = 300 are provided.     

The uniform Korn–Poincaré inequality in thin domains

We study the Korn-Poincaré inequality:

‖u_‖W1,2(Sh)?Ch‖D(u)_‖L²(Sh),     

Asymptotic Properties of Orthogonal Polynomials with Respect to a Non-discrete Jacobi-Sobolev Inner Product

Let {Q _n ^(α,β) (x)} _n=0 ^∞ denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product

$\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)d\mu_{\alpha,\beta}(x)+\lambda\int_{-1}^{1}f'(x)g'(x)d\nu_{\alpha,\beta}(x)$

where λ>0 and d μ _α,β(x)=(x?a)(1?x)^α?1(1+x)^β?1 dx, d ν _α,β(x)=(1?x) ^α (1+x) ^β dx with aα,β>0. Their inner strong asymptotics on (?1,1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Q _n ^(α,β) are obtained.

     

Compactness of Riesz transform commutator associated with Bessel operators

Let λ > 0 and

$${\Delta _\lambda }: = - \frac{{{d^2}}}{{d{x^2}}} - \frac{{2\lambda }}{x}\frac{d}{{dx}}$$

be the Bessel operator on R₊:= (0,∞). We first introduce and obtain an equivalent characterization of CMO(R₊, x^2λdx). By this equivalent characterization and by establishing a new version of the Fréchet-Kolmogorov theorem in the Bessel setting, we further prove that a function b ∈ BMO(R⁺, x^2λdx) is in CMO(R⁺, x^2λdx) if and only if the Riesz transform commutator xxxx is compact on L^p(R⁺, x^2λdx) for all p ∈ (1,∞).

     

On Representation of a Solution to the Cauchy Problem by a Fourier Series in Sobolev-Orthogonal Polynomials Generated by Laguerre Polynomials

We consider the problem of representing a solution to the Cauchy problem for an ordinary differential equation as a Fourier series in polynomials l _r,k ^α (x) (k = 0, 1,...) that are Sobolev-orthonormal with respect to the inner product

$$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{(v)}}(0){g^{(v)}}} (0) + \int\limits_0^\infty {{f^{(r)}}(t)} {g^{(r)}}(t){t^\alpha }{e^{ - t}}dt$$

, and generated by the classical orthogonal Laguerre polynomials L _k ^α (x) (k = 0, 1,...). The polynomials l _r,k ^α (x) are represented as expressions containing the Laguerre polynomials L _n ^α?r (x). An explicit form of the polynomials l _r,k+r ^α (x) is established as an expansion in the powers x ^r+l , l = 0,..., k. These results can be used to study the asymptotic properties of the polynomials l _r,k ^α (x) as k→∞and the approximation properties of the partial sums of Fourier series in these polynomials.

     

Obstructions to the uniform stability of a <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup

Let T _t : X → X be a C ₀-semigroup with generator A. We prove that if the abscissa of uniform boundedness of the resolvent s ₀(A) is greater than zero then for each nondecreasing function h(s): ?₊ → R ₊ there are x′ ∈ X′ and x ∈ X satisfying ∫ ₀ ^∞ h(|〈x′, T _x x〉|)dt = ∞. If i? ∩ Sp(A) ≠ Ø then such x may be taken in D(A ^∞).     

Generalized derivations acting on multilinear polynomials in prime rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x₁,..., x _n ) a multilinear polynomial over C which is not central valued on R. If

$$F(f(r))G(f(r)) = H(f(r)^2 )$$

for all r = (r₁,..., r _n ) ∈ I ⁿ , then one of the following conditions holds:

1. (1)
   there exist a ∈ C and b ∈ U such that F(x) = ax, G(x) = xb and H(x) = xab for all x ∈ R
    
2. (2)
   there exist a, b ∈ U such that F(x) = xa, G(x) = bx and H(x) = abx for all x ∈ R, with ab ∈ C
    
3. (3)
   there exist b ∈ C and a ∈ U such that F(x) = ax, G(x) = bx and H(x) = abx for all x ∈ R
    
4. (4)
   f(x₁,..., x _n )² is central valued on R and one of the following conditions holds

   1. (a)
      there exist a, b, p, p’ ∈ U such that F(x) = ax, G(x) = xb and H(x) = px + xp’ for all x ∈ R, with ab = p + p’
       
   2. (b)
      there exist a, b, p, p’ ∈ U such that F(x) = xa, G(x) = bx and H(x) = px + xp’ for all x ∈ R, with p + p’ = ab ∈ C.
       
    

     

Norms on L of Periodic Interpolation Splines with Equidistant Nodes

We consider the set S _r,n of periodic (with period 1) splines of degree r with deficiency 1 whose nodes are at n equidistant points x_i=i / n. For n-tuples y = (y₀, ... , y_n-1), we take splines s _r,n (y, x) from S _r,n solving the interpolation problem

$$s_{r,n} (y,t_i ) = y_i,$$

where t _i = x _i if r is odd and t _i is the middle of the closed interval [x _i , x _i+1 ] if r is even. For the norms L _r,n ^* of the operator y → s _r,n (y, x) treated as an operator from l¹ to L¹ [0, 1] we establish the estimate

$$L_{r,n}^ * = \frac{4}{{\pi ^2 n}}log min(r,n) + O\left( {\frac{1}{n}} \right)$$

with an absolute constant in the remainder. We study the relationship between the norms L _r,n ^* and the norms of similar operators for nonperiodic splines.

     

Stochastic Hamiltonian flows with singular coefficients

In this paper, we study the following stochastic Hamiltonian system in ?^2d (a second order stochastic differential equation):

$$d{\dot X_t} = b({X_t},{\dot X_t})dt + \sigma ({X_t},{\dot X_t})d{W_t},({X_0},{\dot X_0}) = (x,v) \in \mathbb{R}^{2d},$$

where b(x; v) : ?^2d → ?^d and σ(x; v): ?^2d → ?^d ? ?^d are two Borel measurable functions. We show that if σ is bounded and uniformly non-degenerate, and b ∈ H _p ^2/3,0 and ?σ ∈ L^p for some p > 2(2d+1), where H _p ^{α, β} is the Bessel potential space with differentiability indices α in x and β in v, then the above stochastic equation admits a unique strong solution so that (x, v) ? Z_t(x, v) := (X_t, ?_t)(x, v) forms a stochastic homeomorphism flow, and (x, v) ? Z_t(x, v) is weakly differentiable with ess.sup_{x, v} E(sup_{t∈[0, T]} |?Z_t(x, v)|^q) < ∞ for all q ? 1 and T ? 0. Moreover, we also show the uniqueness of probability measure-valued solutions for kinetic Fokker-Planck equations with rough coefficients by showing the well-posedness of the associated martingale problem and using the superposition principle established by Figalli (2008) and Trevisan (2016).