The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

We study a variant of the Whitney extension problem (1934) for the space . We identify with a space of Lipschitz mappings from into the space of polynomial fields on equipped with a certain metric. This identification allows us to reformulate the Whitney problem for as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of . We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for due to C. Fefferman.

     

Helicoidal minimal surfaces of prescribed genus

For every genus g, we prove that \({\mathbf{S}^2\times\mathbf{R}}\) contains complete, properly embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the \({\mathbf{S}^2}\) tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in \({\mathbf{R}^3}\) that are helicoidal at infinity. We prove that helicoidal surfaces in \({\mathbf{R}^3}\) of every prescribed genus occur as such limits of examples in \({\mathbf{S}^2\times\mathbf{R}}\).     

Searching for a Best Least Absolute Deviations Solution of an Overdetermined System of Linear Equations Motivated by Searching for a Best Least Absolute Deviations Hyperplane on the Basis of Given Data

We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations Xa=z, X∈?^m×n, m≥n, \(\mathbf{a}\in \mathbb{R}^{n}, \mathbf {z}\in\mathbb{R}^{m}\). This problem is equivalent to the problem of determining a best LAD-hyperplane x?a ^T x, x∈? ⁿ on the basis of given data \((\mathbf{x}_{i},z_{i}), \mathbf{x}_{i}= (x_{1}^{(i)},\ldots,x_{n}^{(i)})^{T}\in \mathbb{R}^{n}, z_{i}\in\mathbb{R}, i=1,\ldots,m\), whereby the minimizing functional is of the form

$F(\mathbf{a})=\|\mathbf{z}-\mathbf{Xa}\|_1=\sum_{i=1}^m|z_i-\mathbf {a}^T\mathbf{x}_i|.$

An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional F is convex, and therefore the point a ^?∈? ⁿ is the point of a global minimum of the functional F if and only if 0∈?F(a ^?).

Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane (x,y)?αx+βy, passing through the origin of the coordinate system, on the basis of the data (x _i ,y _i ,z _i ),i=1,…,m.     

Codes with the identifiable parent property for multimedia fingerprinting

Let \({\mathcal {C}}\) be a q-ary code of length n and size M, and \({\mathcal {C}}(i) = \{\mathbf{c}(i) \ | \ \mathbf{c}=(\mathbf{c}(1), \mathbf{c}(2), \ldots , \mathbf{c}(n))^{T} \in {\mathcal {C}}\}\) be the set of ith coordinates of \({\mathcal {C}}\). The descendant code of a sub-code \({\mathcal {C}}^{'} \subseteq {\mathcal {C}}\) is defined to be \({\mathcal {C}}^{'}(1) \times {\mathcal {C}}^{'}(2) \times \cdots \times {\mathcal {C}}^{'}(n)\). In this paper, we introduce a multimedia analogue of codes with the identifiable parent property (IPP), called multimedia IPP codes or t-MIPPC(n, M, q), so that given the descendant code of any sub-code \({\mathcal {C}}^{'}\) of a multimedia t-IPP code \({\mathcal {C}}\), one can always identify, as IPP codes do in the generic digital scenario, at least one codeword in \({\mathcal {C}}^{'}\). We first derive a general upper bound on the size M of a multimedia t-IPP code, and then investigate multimedia 3-IPP codes in more detail. We characterize a multimedia 3-IPP code of length 2 in terms of a bipartite graph and a generalized packing, respectively. By means of these combinatorial characterizations, we further derive a tight upper bound on the size of a multimedia 3-IPP code of length 2, and construct several infinite families of (asymptotically) optimal multimedia 3-IPP codes of length 2.     

Some 0/1 polytopes need exponential size extended formulations

We prove that there are 0/1 polytopes ${P \subseteq \mathbb{R}^{n}}$ that do not admit a compact LP formulation. More precisely we show that for every n there is a set ${X \subseteq \{ 0,1\}^n}$ such that conv(X) must have extension complexity at least ${2^{n/2\cdot(1-o(1))}}$ . In other words, every polyhedron Q that can be linearly projected on conv(X) must have exponentially many facets. In fact, the same result also applies if conv(X) is restricted to be a matroid polytope. Conditioning on ${\mathbf{NP}\not\subseteq \mathbf{P_{/poly}}}$ , our result rules out the existence of a compact formulation for any ${\mathbf{NP}}$ -hard optimization problem even if the formulation may contain arbitrary real numbers.     

An Example of an Unbounded Maximal Singular Operator

Suppose that an even integrable function Ω on the unit sphere S ¹ in R ² with mean value zero satisfies

$\mathop{\mathrm{essup}}\limits_{\xi\in \mathbf{S}^{1}}\biggl|\int_{\mathbf{S}^{1}}\Omega(\theta)\log\frac{1}{|\theta\cdot\xi|}\,d\theta\biggr|<+\infty,$

then the singular integral operator T _Ω given by convolution with the distribution p.v.?Ω(x/|x|)|x|^?2, initially defined on Schwartz functions, extends to an L ²-bounded operator. We construct examples of a function Ω satisfying the above conditions and of a continuous bounded integrable function f such that

$\limsup_{\epsilon\to 0^+}\biggl|\int_{\epsilon<|y|}\Omega(y/|y|)|y|^{-2}f(x-y)dy\biggr|=\infty\quad \hbox{a. e.}$

     

More,or less,trivial matrix functional equations

Various functional equations satisfied by one or two (N × N)-matrices \({\mathbf{F}(z) }\) and \({\mathbf{G}(z) }\) depending on the scalar variable z are investigated, with N an arbitrary positive integer. Some of these functional equations are generalizations to the matrix case (N > 1) of well-known functional equations valid in the scalar (N = 1) case, such as \({\mathbf{F}(x) \, \mathbf{F}(y) = \, \mathbf{F}(x y) \, \rm and \, \mathbf{G}({\it x}) \, \mathbf{G}({\it y}) = \mathbf{G}({\it x+y}) }\); others—such as \({\mathbf{G}(y) \, \mathbf{F}(x) = \mathbf{F}(x) \, \mathbf{G}(xy) }\)—possess nontrivial solutions only in the matrix case (N > 1), namely their scalar (N = 1) counterparts only feature quite trivial solutions. It is also pointed out that if two (N × N)-matrices \({\mathbf{F}(x) \, \rm and \, \mathbf{G}({\it y})}\) satisfy the triplet of functional equations written above—and nontrivial examples of such matrices are exhibited— then they also satisfy an endless hierarchy of matrix functional relations involving an increasing number of scalar independent variables, the first items of which read \({\mathbf{F}(x_{1}) \, \mathbf{G}(y_{1}) \, \mathbf{F} (x_{2}) = \mathbf{F}(x_{1} x_{2}) \, \mathbf{G } (x_{2} y_{1}) \, \rm and \, \mathbf{G}({\it y}_{1}) \, \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it y}_{2}) = \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it x}_{1} {\it y}_{1}+{\it y}_{2}) }\).     

Word Problem of the Perkins Semigroup via Directed Acyclic Graphs

For a word w in an alphabet Γ, the alternation word digraph Alt(w), a certain directed acyclic graph associated with w, is presented as a means to analyze the free spectrum of the Perkins monoid \(\mathbf{B_2^1}\). Let \((f_n^{\mathbf{B_2^1}})\) denote the free spectrum of \(\mathbf{B_2^1}\), let a _n be the number of distinct alternation word digraphs on words whose alphabet is contained in {x ₁,..., x _n }, and let p _n denote the number of distinct labeled posets on {1,..., n}. The word problem for the Perkins semigroup \(\mathbf{B_2^1}\) is solved here in terms of alternation word digraphs: Roughly speaking, two words u and v are equivalent over \(\mathbf{B_2^1}\) if and only if certain alternation graphs associated with u and v are equal. This solution provides the main application, the bounds: \(p_n \leq a_n \leq f_n^{\mathbf{B_2^1}} \leq 2^{n}a_{2n}^2\). A result of the second author in a companion paper states that \((\operatorname{log} \; a_n)\in O(n^3)\), from which it follows that \((\operatorname{log} f_n^{\mathbf{B_2^1}})\in O(n^3)\) as well. Alternation word digraphs are of independent interest combinatorially. It is shown here that the computational complexity problem that has as instance {u,v} where u,v are words of finite length, and question “Is Alt(u)?=?Alt(v)?”, is co-NP-complete. Additionally, alternation word digraphs are acyclic, and certain of them are natural extensions of posets; each realizer of a finite poset determines an extension by an alternation word digraph.     

Min-max and robust polynomial optimization

We consider the robust (or min-max) optimization problem

$J^*:=\max_{\mathbf{y}\in{\Omega}}\min_{\mathbf{x}}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in\mathbf{\Delta}\}$

where f is a polynomial and \({\mathbf{\Delta}\subset\mathbb{R}^n\times\mathbb{R}^p}\) as well as \({{\Omega}\subset\mathbb{R}^p}\) are compact basic semi-algebraic sets. We first provide a sequence of polynomial lower approximations \({(J_i)\subset\mathbb{R}[\mathbf{y}]}\) of the optimal value function \({J(\mathbf{y}):=\min_\mathbf{x}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in \mathbf{\Delta}\}}\). The polynomial \({J_i\in\mathbb{R}[\mathbf{y}]}\) is obtained from an optimal (or nearly optimal) solution of a semidefinite program, the ith in the “joint + marginal” hierarchy of semidefinite relaxations associated with the parametric optimization problem \({\mathbf{y}\mapsto J(\mathbf{y})}\), recently proposed in Lasserre (SIAM J Optim 20, 1995-2022, 2010). Then for fixed i, we consider the polynomial optimization problem \({J^*_i:=\max\nolimits_{\mathbf{y}}\{J_i(\mathbf{y}):\mathbf{y}\in{\Omega}\}}\) and prove that \({\hat{J}^*_i(:=\displaystyle\max\nolimits_{\ell=1,\ldots,i}J^*_\ell)}\) converges to J* as i → ∞. Finally, for fixed ? ≤ i, each \({J^*_\ell}\) (and hence \({\hat{J}^*_i}\)) can be approximated by solving a hierarchy of semidefinite relaxations as already described in Lasserre (SIAM J Optim 11, 796–817, 2001; Moments, Positive Polynomials and Their Applications. Imperial College Press, London 2009).

     

Normal curvature of CR submanifolds of maximal CR dimension of the complex projective space

We prove that there do not exist CR submanifolds Mⁿ of maximal CR dimension of a complex projective space \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) with flat normal connection D of M, when the distinguished normal vector field is parallel with respect to D. If D is lift-flat, then there exists a totally geodesic complex projective subspace \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\) of \({\mathbf{P}^{\frac{n+p}{2}}(\mathbf{C})}\) such that M is a real hypersurface of \({\mathbf{P}^{\frac{n+1}{2}}(\mathbf{C})}\).     

Mean Square Estimates for Coefficients of Symmetric Power <Emphasis Type="Italic">L</Emphasis>-Functions

Let L(sym ^j f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that

$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$

and

$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$

     

A Refinement of the Kolmogorov–Marcinkiewicz–Zygmund Strong Law of Large Numbers

Let {X _n ; n≥1} be a sequence of independent copies of a real-valued random variable X and set S _n =X ₁+???+X _n , n≥1. This paper is devoted to a refinement of the classical Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers. We show that for 0<p<2,

$\sum_{n=1}^{\infty}\frac{1}{n}\biggl(\frac{|S_{n}|}{n^{1/p}}\biggr)<\infty\quad \mbox{almost surely}$

if and only if

$\begin{cases}\mathbb{E}|X|^{p}<\infty &; \mbox{if }0 < p < 1,\\\mathbb{E}X=0,\ \sum_{n=1}^{\infty}\frac{|\mathbb{E}XI\{|X|\leq n\}|}{n}<\infty,\mbox{ and }\\\sum_{n=1}^{\infty}\frac{\int_{\min\{u_{n},n\}}^{n}\mathbb{P}(|X|>t)\,dt}{n}<\infty &; \mbox{if }p = 1,\\\mathbb{E}X=0\mbox{ and }\int_{0}^{\infty}\mathbb{P}^{1/p}(|X|>t)\,dt<\infty,&;\mbox{if }1 < p < 2,\end{cases}$

where \(u_{n}=\inf \{t:~\mathbb{P}(|X|>t)<\frac{1}{n}\}\), n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159–186, 1974) inequality to obtain some general results for sums of the form \(\sum_{n=1}^{\infty}a_{n}\|\sum_{i=1}^{n}V_{i}\|\) (where {V _n ; n≥1} is a sequence of independent Banach-space-valued random variables, and a _n ≥0, n≥1), which may be of independent interest, but which we apply to \(\sum_{n=1}^{\infty}\frac{1}{n}(\frac{|S_{n}|}{n^{1/p}})\).

     

A general low-order partial regularity theory for asymptotically convex functionals with asymptotic dependence on the minimizer

We consider a minimizer \(u\in W^{1,p}(\Omega )\), where \(\Omega \subseteq \mathbb {R}^{n}\) is a bounded open set, of the integral functional

$$\begin{aligned} u\mapsto \int _{\Omega }f(x,u,Du)\ dx \end{aligned}$$

in the case when \(f : \Omega \subseteq \mathbb {R}^{n}\times \mathbb {R}^{N}\times \mathbb {R}^{N\times n}\rightarrow \mathbb {R}\) is asymptotically related to a more regular function; since we assume that \(N\ge 2\), we study here the vectorial case. The asymptotical relatedness condition is such that dependence on u is allowed even as \(|\xi |\rightarrow +\infty \). Unlike in previous work f is allowed to satisfy a more general growth condition, which permits a coupling between the x and u variables. Due to the generality of the asymptotical relatedness condition this coupling induces a subtle restriction on the regularity required of the various functions in the growth condition, and we study these restrictions in this paper. In addition, here we do not obtain the full spectrum of Hölder continuity for the minimizer, but rather a restricted range of Hölder exponents that depend on the initial data.

     

Polynomiality of some hook-length statistics

We prove a conjecture of Okada giving an exact formula for a certain statistic for hook-lengths of partitions:

$\frac{1}{n!} \sum_{\lambda \vdash n} f_{\lambda}^2 \sum_{u \in \lambda} \prod_{i=1}^{r}\bigl(h_u^2 - i^2\bigr) = \frac{1}{2(r+1)^2} \binom{2r}{r}\binom{2r+2}{ r+1} \prod_{j=0}^{r} (n-j),$

where f _λ is the number of standard Young tableaux of shape λ and h _u is the hook length of the square u of the Young diagram of λ. We also obtain other similar formulas.

     

Hrmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters

The main purpose of this paper is to establish the Hormander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k≥ 3 using the multiparameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k≥ 3:■where x =(x_1,x_2,x_3)∈R~(n_1)×R~(n_2)×R~(n_3) and ξ =(ξ_1,ξ_2,ξ_3)∈R~(n_1)×R~(n_2)×R~(n_3). One of our main results is the following:Assume that m(ξ) is a function on R~(n_1+n_2+n_3) satisfying ■ with s_i n_i(1/p-1/2) for 1≤i≤3. Then T_m is bounded from H~p(R~(n_1)×R~(n_2)×R~(n_3) to H~p(R~(n_1)×R~(n_2)×R~(n_3)for all 0 p≤1 and ■ Moreover, the smoothness assumption on s_i for 1≤i≤3 is optimal. Here we have used the notations m_(j,k,l)(ξ)=m(2~jξ_1,2~kξ_2,2~lξ_3)Ψ(ξ_1)Ψ(ξ_2)Ψ(ξ_3) and Ψ(ξ_i) is a suitable cut-off function on R~(n_i) for1≤i≤3, and W~(s_1,s_2,s_3) is a three-parameter Sobolev space on R~(n_1)×R~(n_2)× R~(n_3).Because the Fefferman criterion breaks down in three parameters or more, we consider the L~p boundedness of the Littlewood-Paley square function of T_mf to establish its boundedness on the multi-parameter Hardy spaces.     

Gabor (super)frames with Hermite functions

We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions H _n . Let h = (H ₀, H ₁, . . . , H _n ) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices \({\Lambda \subseteq \mathbb{R} ^2}\) such that the Gabor system \({\{ {\rm e}^{2\pi i \lambda _{2} t}{\bf h} (t-\lambda _1): \lambda = (\lambda _1, \lambda _2) \in \Lambda \}}\) is a frame for \({L^2 (\mathbb{R} , \mathbb{C} ^{n+1})}\). As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor frame and a new estimate for the lower frame bound. The main tools are growth estimates for the Weierstrass σ-function, a new type of interpolation problem for entire functions on the Bargmann–Fock space, and structural results about vector-valued Gabor frames.     

Complete real Kähler submanifolds in codimension two

Minimal isometric immersions \(f : M^{2n} \rightarrow {\mathbb{R}}^{2n+2}\) in codimension two from a complete Kähler manifold into Euclidean space had been classified in Dajczer and Gromoll (Invent Math 119:235–242, 1995) for n ≥  3. In this note we describe the non-minimal situation showing that, if f is real analytic but not everywhere minimal, then f is a cylinder over a real Kähler surface \(g : N^4 \rightarrow {\mathbb{R}}^6\) , that is, \(M^{2n} = N^4 \times {\mathbb{C}}^{n-2}\) and f = g × id split, where \({id} : {\mathbb{C}}^{n-2} \cong {\mathbb{R}}^{2n-4}\) is the identity map. Moreover, g can be further described.     

Hyperbolas over two-dimensional Fibonacci quasilattices

For the number n _s (α, β; X) of points (x ₁ , x ₂) in the two-dimensional Fibonacci quasilattices \( \mathcal{F}_m^2 \) of level m?=?0, 1, 2,… lying on the hyperbola x ₁ ² ? ??αx ₂ ² ?=?β and such that 0?≤?x ₁? ≤?X, x ₂? ≥?0, the asymptotic formula

$ {n_s}\left( {\alpha, \beta; X} \right)\sim {c_s}\left( {\alpha, \beta } \right)\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty $

is established, and the coefficient c _s (α, β) is calculated exactly. Using this, we obtain the following result. Let F _m be the Fibonacci numbers, A _i ∈ \( \mathbb{N} \), i?=?1, 2, and let \( \overleftarrow {{A_i}} \) be the shift of A _i in the Fibonacci numeral system. Then the number n _s (X) of all solutions (A ₁ , A ₂) of the Diophantine system

$ \left\{ {\begin{array}{*{20}{c}} {A_1^2 + \overleftarrow {A_1^2} - 2{A_2}{{\overleftarrow A }_2} + \overleftarrow {A_2^2} = {F_{2s}},} \\ {\overleftarrow {A_1^2} - 2{A_1}{{\overleftarrow A }_1} + A_2^2 - 2{A_2}{{\overleftarrow A }_2} + 2\overleftarrow {A_2^2} = {F_{2s - 1}},} \\ \end{array} } \right. $

0?≤?A ₁? ≤?X, A ₂? ≥?0, satisfies the asymptotic formula

$ {n_s}(X)\sim \frac{{{c_s}}}{{{\text{ar}}\cosh \left( {{{1} \left/ {\tau } \right.}} \right)}}\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty . $

Here τ?=?(?1?+?√5)/2 is the golden ratio, and c _s ?=?1/2 or 1 for s?=?0 or s?≥?1, respectively.

     

A solution to the Bézout equation in <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">K</Emphasis>) without Gelfand theory

Let \({M\subseteq \mathbb{C}}\) be compact and \({K\subseteq M}\) closed, and let A(K, M) be the uniform algebra of all functions continuous on M and holomorphic in the interior K° of K. We present a constructive proof of Arens’ classical result that for \({(f_{1},\ldots,f_{n})\in A(K,M)^{n}}\) the Bézout equation \({\sum_{j=1}^{n} a_{j}f_{j}=1}\) has a solution in A(K, M) if and only if the functions f _j have no common zero in K. We shall also consider matrix-valued Bézout equations.     

Cardinal interpolation with general multiquadrics

This paper studies the cardinal interpolation operators associated with the general multiquadrics, ? _{α, c} (x)=(∥x∥² + c ²) ^α , \(x\in \mathbb {R}^{d}\). These operators take the form

$$\mathcal{I}_{\alpha,c}\mathbf{y}(x) = \sum\limits_{j\in\mathbb{Z}^{d}}y_{j}L_{\alpha,c}(x-j),\quad\mathbf{y}=(y_{j})_{j\in\mathbb{Z}^{d}},\quad x\in\mathbb{R}^{d}, $$

where L _{α, c} is a fundamental function formed by integer translates of ? _{α, c} which satisfies the interpolatory condition \(L_{\alpha ,c}(k) = \delta _{0,k},\; k\in \mathbb {Z}^{d}\). We consider recovery results for interpolation of bandlimited functions in higher dimensions by limiting the parameter \(c\to \infty \). In the univariate case, we consider the norm of the operator \(\mathcal {I}_{\alpha ,c}\) acting on ? _p spaces as well as prove decay rates for L _{α, c} using a detailed analysis of the derivatives of its Fourier transform, \(\widehat {L_{\alpha ,c}}\).