Nil-Manifolds Cannot be Immersed as Hypersurfaces in Euclidean Spaces

We prove that the $2n + 1$ -dimensional Heisenberg group H _n and the 4-manifolds $Nil^4 $ and $Nil^3 \times \mathbb{R}$ endowed with an arbitrary left-invariant metric admit no C ³-regular immersions into Euclidean spaces $\mathbb{R}^{2n + 2} $ and $\mathbb{R}^5 $ , respectively.     

Codimension 1 Subvarieties of {\mathcal{M}_g } and Real Gonality of Real Curves

Let ${\mathcal{M}_g }$ be the moduli space of smooth complex projective curves of genus g. Here we prove that the subset of ${\mathcal{M}_g }$ formed by all curves for which some Brill-Noether locus has dimension larger than the expected one has codimension at least two in ${\mathcal{M}_g }$ . As an application we show that if ${X \in \mathcal{M}_g }$ is defined over $\mathbb{R}$ then there exists a low degree pencil ${u:X \to \mathbb{P}^1 }$ defined over $\mathbb{R}.$      

Approximation by M. Riesz Kernels in L p for p<1

Let α > 0. We consider the linear span $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right)$ of scalar Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^\alpha }}} \right\}_{a \in \mathbb{R}^n }$ and the linear span $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right)$ of vector Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^{\alpha + 1} }}\left( {x - a} \right)} \right\}_{a \in \mathbb{R}^n }$ . We study the following problems. (1) When is the intersection $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n } \right)$ dense in L^p(?ⁿ)? (2) When is the intersection $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n ,\mathbb{R}^n } \right)$ dense in L^p(?ⁿ, ?ⁿ)? Bibliography: 15 titles.     

On 1-isometries of affine quadrics over finite fields

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume $4 \leqslant dim V \leqslant \infty \wedge |\mathbb{F}| \in \mathbb{N}$ . A 1-isometry of the central quadric $\mathcal{F}: = \{ x \in V|q(x) = 1\}$ is a permutation ? of $\mathcal{F}$ such that (*) $$q(x - y) = \nu \Leftrightarrow q(x^\varphi - y^\varphi ) = \nu \forall x,y \in \mathcal{F}$$ holds true for a fixed element ν of $\mathbb{F}$ . For arbitraryν ∈ $\mathbb{F}$ we prove that? is induced (in a certain sense) by a semi-linear bijection $(\sigma ,\varrho ):(V,\mathbb{F}) \to (V,\mathbb{F})$ such thatq oσ =? oq, provided $\mathcal{F}$ contains lines and the exceptional case $(\nu = 2 \Lambda |\mathbb{F}| = 3 \Lambda \dim V = 4 \Lambda |\mathcal{F}| = 24)$ is excluded. In the exceptional case and as well in case of dim V = 3 there are counterexamples. The casesν ≠ 2 and v=2 require different techniques.     

On permutations of central quadrics which preserve an inner distance

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume dimV ≥ 4 and ¦ $\mathbb{F}$ ¦ ≥ 4. We consider a permutation ? of the central affine quadric $\mathcal{F}$ := {x εV ¦q(x) = 1} such that $$(*)x \cdot y = \mu \Leftrightarrow x^\varphi \cdot y^\varphi = \mu \forall x,y\varepsilon \mathcal{F}$$ holds true, where μ is a fixed element of $\mathbb{F}$ and where “·” is the scalar product associated withq. We prove that ? is induced (in a certain sense) by a semi-linear bijection (σ,?): (V, $\mathbb{F}$ ) → (V, $\mathbb{F}$ ) such thatq o ?o q, provided $\mathcal{F}$ contains lines and the pair (μ, $\mathbb{F}$ ) has additional properties if there ar no planes in $\mathcal{F}$ . The cases μ,≠ 0 and μ = 0 require different techniques.     

The Quintic Grassmannian G_{1,4,2} in PG(9, 2)

The 155 points of the Grassmannian $G_{1,4,2}$ of lines of PG (4, 2) = $\mathbb{P}V\left( {5,2} \right)$ are those points $x \in {\text{PG}}\left( {{\text{9,2}}} \right) = \mathbb{P}\left( { \wedge {}^2V\left( {5,2} \right)} \right)$ which satisfy a certain quintic equation Q(x) = 0. (The quintic polynomial Q is given explicitly in Shaw and Gordon [3].) A projective flat X $ \subset $ PG (9, 2) will be termed odd or even according as X intersects $G_{1,4,2}$ in an odd or even number of points. Let $Q^\ddag \left( {x_1 ,...,x_5 } \right)$ denote the alternating quinquelinear form obtained by completely polarizing Q. We define the associate Y = X ^# of a r-flat X $ \subset $ PG (9, 2) by $$Y = \left\{ {y \in {\text{PG}}\left( {n{\text{,2}}} \right)\left| {Q^\ddag \left( {x_1 ,x_{2,} ,x_3 ,x_4 ,y} \right)} \right. = 0,\quad {\text{for}}\;{\text{all}}\,x_1 ,x_{2,} ,x_3 ,x_4 \in X} \right\}.$$ . Because $Q^\ddag$ is quinquelinear, the associate X ^# of an r-flat X is an s-flat for some s. The cases where r = 4 are of particular interest: if X is an odd 4-flat then X $ \subseteq$ X ^# while if X is an even 4-flat then X ^# is necessarily also a 4-flat which is moreover disjoint from X. We give an example of an odd 4-flat X which is self-associate: X ^# = X. An example of an even 4-flat X such that $\left( {X^\# } \right)^\#$ = X is provided by any 4-flat X which is external to $G_{1,4,2}$ . However, it appears that the two possibilities just illustrated, namely X ^# = X for an odd 4-flat and $\left( {X^\# } \right)^\#$ = X for an even 4-flat, are the exception rather than the rule. Indeed, we provide examples of odd 4-flats for which X ^# = PG (9, 2) and of even 4-flats for which ${X^{\# \# \# } }$ = X.     

Uniqueness of \mathbb{C}^{ * } - and \mathbb{C}_{{\text{ + }}} -actions on Gizatullin Surfaces

A Gizatullin surface is a normal affine surface V over $ \mathbb{C} $ , which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with $ \mathbb{C}_{{\text{ + }}} $ -actions on V considered up to a “speed change”. Non-Gizatullin surfaces are known to admit at most one $ \mathbb{A}^{1} $ -fibration V → S up to an isomorphism of the base S. Moreover, an effective $ \mathbb{C}^{ * } $ -action on them, if it does exist, is unique up to conjugation and inversion t $ \mapsto $ t ^?1 of $ \mathbb{C}^{ * } $ . Obviously, uniqueness of $ \mathbb{C}^{ * } $ -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when $ \mathbb{A}^{{\text{1}}} $ -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base $ S \cong \mathbb{A}^{{\text{1}}} $ . We exhibit as well large subclasses of Gizatullin $ \mathbb{C}^{ * } $ -surfaces for which a $ \mathbb{C}^{ * } $ -action is essentially unique and for which there are at most two conjugacy classes of $ \mathbb{A}^{{\text{1}}} $ -fibrations over $ \mathbb{A}^{{\text{1}}} $ .     

Subexponential loss rates in a GI/GI/1 queue with applications

Consider a single server queue with i.i.d. arrival and service processes, $\{ A,A_n ,n \geqslant 0\} $ and $\{ C,\;C_n ,n\;\; \geqslant \;\;0\} $ , respectively, and a finite buffer B. The queue content process $\{ Q_n^B ,n \geqslant 0\} $ is recursively defined as $Q_{n + 1}^B = \min ((Q_n^B + A_{n + 1} - C_{n + 1} )^ + ,B),\;\;q^ + = \max (0,q)$ . When $\mathbb{E}(A - C) < 0$ , and A has a subexponential distribution, we show that the stationary expected loss rate for this queue $E(Q_n^B + A_{n + 1} - C_{n + 1} - B)^ + $ has the following explicit asymptotic characterization: $${\mathbb{E}}\left( {Q_n^B + A_{n + 1} - C_{n + 1} - B} \right)^ + ~{\mathbb{E}}\left( {A - B} \right)^ + {as} B \to \infty ,$$ independently of the server process C _n . For a fluid queue with capacity c, M/G/∞ arrival process A _t , characterized by intermediately regularly varying on periods σ^on, which arrive with Poisson rate Λ, the average loss rate $\lambda _{{loss}}^B $ satisfies λ _loss ^B ～ Λ E(τ^onη — B)⁺ as B → ∞, where $\eta = r + \rho - c,\;\rho \; = \mathbb{E}A_t < \;\;c;r\;\;(c \leqslant r)$ is the rate at which the fluid is arriving during an on period. Accuracy of the above asymptotic relations is verified with extensive numerical and simulation experiments. These explicit formulas have potential application in designing communication networks that will carry traffic with long-tailed characteristics, e.g., Internet data services.     

The Hilbert functions ofk-configurations in\mathbb{P}^2 and\mathbb{P}^3

In this paper, we proved the set of points which are the vertices of then-gon in $\mathbb{P}^2 $ (n ≥ 3) has the Uniform Position Property and what the graded free resolutions of the ideals ofk-configurations in $\mathbb{P}^3 $ are.     

The index of centralizers of elements in classical Lie algebras

The index of a finite-dimensional Lie algebra $\mathfrak{g}$ is the minimum of dimensions of the stabilizers $\mathfrak{g}_\alpha $ over all covectors $\alpha \in \mathfrak{g}^ * $ . Let $\mathfrak{g}$ be a reductive Lie algebra over a field $\mathbb{K}$ of characteristic ≠ = 2. Élashvili conjectured that the index of $\mathfrak{g}_\alpha $ is always equal to the index, or, which is the same, the rank of $\mathfrak{g}$ . In this article, Élashvili’s conjecture is proved for classical Lie algebras. Furthermore, it is shown that if $\mathfrak{g} = \mathfrak{g}\mathfrak{l}_n $ or $\mathfrak{g} = \mathfrak{s}\mathfrak{p}_{2n} $ and $e \in \mathfrak{g}$ is a nilpotent element, then the coadjoint action of $\mathfrak{g}_e $ has a generic stabilizer. For $\mathfrak{g}$ , we give examples of nilpotent elements $e \in \mathfrak{g}$ such that the coadjoint action of $\mathfrak{g}_e $ does not have a generic stabilizer.     

Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps

We prove a global implicit function theorem. In particular we show that any Lipschitz map ${f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}$ (with n-dim. image) can be precomposed with a bi-Lipschitz map ${\bar{g} : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n} \times \mathbb{R}^{m}}$ such that ${f \circ \bar{g}}$ will satisfy, when we restrict to a large portion of the domain ${E \subset \mathbb{R}^{n} \times \mathbb{R}^{m}}$ , that ${f \circ \bar{g}}$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map ${\bar{g}}$ distorts ${\mathbb{R}^{n+m}}$ in a controlled manner so that the fibers of f are straightened out. Furthermore, our results stay valid when the target space is replaced by any metric space. A main point is that our results are quantitative: the size of the set E on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman’s 1979 construction of a C ¹ map from [0, 1]³ onto [0, 1]² with rank ≤ 1 everywhere. On route we prove an extension theorem which is of independent interest. We show that for any D ≥ n, any Lipschitz function ${f : [0,1]^{n} \rightarrow \mathbb{R}^{D}}$ gives rise to a large (in an appropriate sense) subset ${E \subset [0,1]^{n}}$ such that ${f|_E}$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of ${\mathbb{R}^{n}}$ . This extends results of Jones and David, from 1988. As a simple corollary, we show that n-dimensional Ahlfors–David regular spaces lying in ${\mathbb{R}^{D}}$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in ${\mathbb{R}^{D}}$ . This was previously known only for D ≥ 2n?+?1 by a result of David and Semmes.     

Inscribing a Regular Octahedron in a Three-Dimensional Convex Body with Smooth Boundary

A norm $|| \cdot ||$ and a convex body K with smooth boundary in the standard Euclidean space $\mathbb{R}^3 $ are considered. It is proved that the boundary $\partial $ K of K contains the vertices AA'BB'CC' of a regular octahedron with $||AA\prime || = ||BB\prime || \geqslant ||CC\prime ||$ (or, optionally, $||AA\prime || = ||BB\prime || \leqslant ||CC\prime ||$ ). Bibliography: 4 titles.     

Uniqueness of integrable solutions to {\nabla \zeta=G \zeta, \zeta|_\Gamma = 0} for integrable tensor coefficients G and applications to elasticity

Let ${\Omega \subset \mathbb{R}^{N}}$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ${\partial\Omega}$ . We show that the solution to the linear first-order system $$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$ is unique if ${G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}$ and ${\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}$ . As a consequence, we prove $$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$ to be a norm for ${P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}$ with Curl ${P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}$ , Curl ${P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}$ for some p, q > 1 with 1/p + 1/q = 1 as well as det ${P \geq c^+ > 0}$ . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ${\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}$ satisfy sym ${(\nabla\Phi^\top\nabla\Psi) = 0}$ for some ${\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}$ with det ${\nabla\Psi \geq c^+ > 0}$ . Then, there exist a constant translation vector ${a \in \mathbb{R}^{3}}$ and a constant skew-symmetric matrix ${A \in \mathfrak{so}(3)}$ , such that ${\Phi = A\Psi + a}$ .     

On Uniform Approximation in R2 of Continuous Functions of Two Variables with Bounded Variation in the Sense of Hardy

Introduce the notation: $\mathbb{Z}$ is the set of integers, $\bar {\mathbb{Z}}={\mathbb{Z}} \cup \{-\infty, +\infty\},{\mathbb{R}}_+^2 =\{x=(x_1,x_2) \in {\mathbb{R}}^2; x_1>0,x_2>0\}$ , $g_{k,m} (x,\alpha,h)= \int\limits_0^1 {g_1 (\frac{(k+u)h_1 - x_1}{\alpha_1})g_2(\frac{(m+u)h_2 - x_2}{\alpha_2})}du$ , where $g_i :\mathbb{R} \to \mathbb{R},x \in \mathbb{R}^2 ,\alpha ,h \in \mathbb{R}_ + ^2 $ . Under certain conditions on the functions g ₁, g ₂, we prove that the system of functions $g_{k,m} (x,\alpha^(n), h^(n)) (k,m \in \bar {\mathbb{Z}})$ , where $\alpha ^{\left( n \right)} ,h^{\left( n \right)} \in \mathbb{R}_ + ^2 $ are arbitrary infinitesimal sequences, is complete in the space C $\mathbb{R}^2 $ of uniformly continuous bounded functions f equipped with the norm $||f|| = \mathop {\sup }\limits_{x \in \mathbb{R}^2 } |f(x)|$ . Starting with the functions g _k,m , it is possible to construct a method for uniform approximating in $\mathbb{R}^2 $ any continuous function of bounded variation in the sense of Hardy. An error estimate is derived in terms of the second order moduli of continuity. Based on the obtained results, we discuss in detail the accuracy of uniform approximation of functions of several variables by linear functions. The error estimates are derived by using second order moduli of continuity. We pay a particular attention to sharpness of constants. Bibliography: 8 titles.     

Sets Computing the Symmetric Tensor Rank

Let ${\nu_{d} : \mathbb{P}^{r} \rightarrow \mathbb{P}^{N}, N := \left( \begin{array}{ll} r + d \\ \,\,\,\,\,\, r \end{array} \right)- 1,}$ denote the degree d Veronese embedding of ${\mathbb{P}^{r}}$ . For any ${P\, \in \, \mathbb{P}^{N}}$ , the symmetric tensor rank sr(P) is the minimal cardinality of a set ${\mathcal{S} \subset \nu_{d}(\mathbb{P}^{r})}$ spanning P. Let ${\mathcal{S}(P)}$ be the set of all ${A \subset \mathbb{P}^{r}}$ such that ${\nu_{d}(A)}$ computes sr(P). Here we classify all ${P \,\in\, \mathbb{P}^{n}}$ such that sr(P) <  3d/2 and sr(P) is computed by at least two subsets of ${\nu_{d}(\mathbb{P}^{r})}$ . For such tensors ${P\, \in\, \mathbb{P}^{N}}$ , we prove that ${\mathcal{S}(P)}$ has no isolated points.     

Best Uniform Rational Approximations of Functions by Orthoprojections

Let C[-1,1] be the Banach space of continuous complex functions $f$ on the interval [-1,1] equipped with the standard maximum norm $\left\| f \right\|$ ; let $\omega \left( \cdot \right) = \omega \left( { \cdot ,f} \right)$ be the modulus of continuity of $f$ ; and let $R_n = R_n \left( f \right)$ be the best uniform approximation of $f$ by rational functions (r.f.) whose degrees do not exceed $n = 1, 2, \ldots $ . The space C[-1,1] is also regarded as a pre-Hilbert space with respect to the inner product given by $\left( {f,g} \right) = \left( {1/\pi } \right)\int_{ - 1}^1 {f\left( x \right)g\left( x \right)} \left( {1 - x^2 } \right)^{ - 1/2} dx$ . Let $z_n = \{ z_1 , z_2 , \ldots z_n \} $ be a set of points located outside the interval [-1,1]. By $F\left( { \cdot ,f,z_n } \right)$ we denote an orthoprojection operator acting from the pre-Hilbert space C[-1,1] onto its ( ${n + 1}$ )-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set $z_n $ . In this paper, we show that if $f$ is not a rational function of degree $ \leqslant n$ , then we can find a set of points $z_n = z_n \left( f \right)$ such that $\left\| {f\left( \cdot \right) - F\left( { \cdot ,f,z_n } \right)} \right\| \leqslant 12R_n ln\frac{3}{{\omega ^{ - 1} \left( {R_n /3} \right)}}.$      

Locally Minimal Trees in n-Normed Spaces

The locally minimal trees in normed spaces $({\mathbb{R}}^2, \rho)$ , where the unit circle ${x \in {\mathbb{R}}^2 | \rho (x)=1}$ in the norm ρ coincides with the regular m-gon (m = 2n) inscribed in the Euclidean unit circle S ¹, are completely classified.     

The Wiener--Hopf Integral Equation in the Supercritical Case

We consider the scalar homogeneous equation $S(x) = \int_0^\infty {K(x - t)S(t)dt,{\text{ }}x \in \mathbb{R}^ + \equiv (0,\infty ),}$ with symmetric kernel $K:K( - x) = K(x),{\text{ }}x \in \mathbb{R}_1$ satisfying the conditions $0 \leqslant K \in L_1 (\mathbb{R}^ + ) \cap C^{\left( 2 \right)} (\mathbb{R}^ + )$ , $\int_0^\infty {K(t)dt > \frac{1}{2}} $ , $K' \leqslant 0{\text{ }}and 0 \leqslant K'' \downarrow {\text{ }}on \mathbb{R}^ + $ . We prove the existence of a real solution S of the equation given above with asymptotic behavior $S(x) = O(x){\text{ as }}x \to + \infty $ .     

Infinite-dimensional quasigroups of finite orders

Let Σ be a finite set of cardinality k > 0, let $\mathbb{A}$ be a finite or infinite set of indices, and let $\mathcal{F} \subseteq \Sigma ^\mathbb{A}$ be a subset consisting of finitely supported families. A function $f:\Sigma ^\mathbb{A} \to \Sigma$ is referred to as an $\mathbb{A}$ -quasigroup (if $\left| \mathbb{A} \right| = n$ , then an n-ary quasigroup) of order k if $f\left( {\bar y} \right) \ne f\left( {\bar z} \right)$ for any ordered families $\bar y$ and $\bar z$ that differ at exactly one position. It is proved that an $\mathbb{A}$ -quasigroup f of order 4 is reducible (representable as a superposition) or semilinear on every coset of $\mathcal{F}$ . It is shown that the quasigroups defined on Σ^?, where ? are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1].     

Postulation of Disjoint Unions of Lines and Double Lines in {\mathbb{P}^3}

A double line ${C \subset \mathbb{P}^3}$ is a connected divisor of type (2, 0) on a smooth quadric surface. Fix ${(a, c) \in \mathbb{N}^2\ \backslash\ \{(0, 0)\}}$ . Let ${X \subset \mathbb{P}^3}$ be a general disjoint union of a lines and c double lines. Then X has maximal rank, i.e. for each ${t \in \mathbb{Z}}$ either ${h^1(\mathcal{I}_X(t)) = 0}$ or ${h^0(\mathcal{I}_X(t)) = 0}$ .