On height isotopy classes of embeddings in the plane of a Morse function of a circle

Let \(\mathrm{SM}_{2n}(S^1,\mathbb {R})\) be a set of stable Morse functions of an oriented circle such that the number of singular points is \(2n\in \mathbb {N}\) and the order of singular values satisfies the particular condition. For an orthogonal projection \(\pi :\mathbb {R}^2\rightarrow \mathbb {R}\), let \({\tilde{f}}_0\) and \({\tilde{f}}_1:S^1\rightarrow \mathbb {R}^2\) be embedding lifts of f. If there is an ambient isotopy \(\tilde{\varphi }_t:\mathbb {R}^2\rightarrow \mathbb {R}^2\) \((t\in [0,1])\) such that \({\pi \circ \tilde{\varphi }}_t(y_1,y_2)=y_1\) and \(\tilde{\varphi }_1\circ {\tilde{f}}_0={\tilde{f}}_1\), we say that \({\tilde{f}}_0\) and \({\tilde{f}}_1\) are height isotopic. We define a function \(I:\mathrm{SM}_{2n}(S^1,\mathbb {R})\rightarrow \mathbb {N}\) as follows: I(f) is the number of height isotopy classes of embeddings such that each rotation number is one. In this paper, we determine the maximal value of the function I equals the n-th Baxter number and the minimal value equals \(2^{n-1}\).     

Fourier Multiplier Theorems Involving Type and Cotype

In this paper we develop the theory of Fourier multiplier operators \(T_{m}:L^{p}({\mathbb R}^{d};X)\rightarrow L^{q}({\mathbb R}^{d};Y)\), for Banach spaces X and Y, \(1\le p\le q\le \infty \) and \(m:{\mathbb R}^d\rightarrow \mathcal {L}(X,Y)\) an operator-valued symbol. The case \(p=q\) has been studied extensively since the 1980s, but far less is known for \(p<q\). In the scalar setting one can deduce results for \(p<q\) from the case \(p=q\). However, in the vector-valued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that X and Y are UMD spaces and that m satisfies a smoothness condition. We show that for \(p<q\) other geometric conditions on X and Y, such as the notions of type and cotype, can be used to study Fourier multipliers. Moreover, we obtain boundedness results for \(T_m\) without any smoothness properties of m. Under smoothness conditions the boundedness results can be extrapolated to other values of p and q as long as \(\tfrac{1}{p}-\tfrac{1}{q}\) remains constant.     

Bending the helicoid

We construct Colding–Minicozzi limit minimal laminations in open domains in \({\mathbb{R}}^3\) with the singular set of C ¹-convergence being any properly embedded C ^1,1-curve. By Meeks’ C ^1,1-regularity theorem, the singular set of convergence of a Colding–Minicozzi limit minimal lamination \({\mathcal{L}}\) is a locally finite collection \(S({\mathcal{L}})\) of C ^1,1-curves that are orthogonal to the leaves of the lamination. Thus, our existence theorem gives a complete answer as to which curves appear as the singular set of a Colding–Minicozzi limit minimal lamination. In the case the curve is the unit circle \({\mathbb{S}}^1(1)\) in the (x ₁, x ₂)-plane, the classical Björling theorem produces an infinite sequence of complete minimal annuli H _n of finite total curvature which contain the circle. The complete minimal surfaces H _n contain embedded compact minimal annuli \(\overline{H}_n\) in closed compact neighborhoods N _n of the circle that converge as \(n \to \infty\) to \(\mathbb {R}^3 - x_3\) -axis. In this case, we prove that the \(\overline{H}_n\) converge on compact sets to the foliation of \(\mathbb {R}^3 - x_3\) -axis by vertical half planes with boundary the x ₃-axis and with \({\mathbb{S}}^1(1)\) as the singular set of C ¹-convergence. The \(\overline{H}_n\) have the appearance of highly spinning helicoids with the circle as their axis and are named bent helicoids.     

Generalized Bernstein Operators on the Classical Polynomial Spaces

We study generalizations of the classical Bernstein operators on the polynomial spaces \(\mathbb {P}_{n}[a,b]\), where instead of fixing \(\mathbf {1}\) and x, we reproduce exactly \(\mathbf {1}\) and a polynomial \(f_1\), strictly increasing on [a, b]. We prove that for sufficiently large n, there always exist generalized Bernstein operators fixing \(\mathbf {1}\) and \(f_1\). These operators are defined by non-decreasing sequences of nodes precisely when \(f_1^\prime > 0\) on (a, b), but even if \(f_1^\prime \) vanishes somewhere inside (a, b), they converge to the identity.     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

On some constacyclic codes over $$\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle $$, their $$\mathbb {Z}_4$$Z4 images,and new codes

In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\).     

Determination of cusp forms by central values of Rankin–Selberg <Emphasis Type="BoldItalic">L</Emphasis>-functions<Superscript>*</Superscript>

Let f be a fixed holomorphic Hecke eigen cusp form of weight k for \( SL\left( {2,{\mathbb Z}} \right) \), and let \( {\mathcal U} = \left\{ {{u_j}:j \geqslant 1} \right\} \) be an orthonormal basis of Hecke–Maass cusp forms for \( SL\left( {2,{\mathbb Z}} \right) \). We prove an asymptotic formula for the twisted first moment of the Rankin–Selberg L-functions \( L\left( {s,f \otimes {u_j}} \right) \) at \( s = \frac{1}{2} \) as u _j runs over \( {\mathcal U} \). It follows that f is uniquely determined by the central values of the family of Rankin–Selberg L-functions \( \left\{ {L\left( {s,f \otimes {u_j}} \right):{u_j} \in {\mathcal U}} \right\} \).     

On behavior of Fourier coefficients by Walsh system

The paper proves that for any ε > 0 there exists ameasurable set E ? [0, 1] with measure |E| > 1 ? ε such that for each f ∈ L¹[0, 1] there is a function \(\tilde f \in {L^1}\left[ {0,1} \right]\) coinciding with f on E whose Fourier-Walsh series converges to \(\tilde f\) in L¹[0, 1]-norm, and the sequence \(\left\{ {\left| {{c_k}\left( {\tilde f} \right)} \right|} \right\}_{n = 0}^\infty \) is monotonically decreasing, where \(\left\{ {{c_k}\left( {\tilde f} \right)} \right\}\) is the sequence of Fourier-Walsh coefficients of \(\left\{ {\left| {{c_k}\left( {\tilde f} \right)} \right|} \right\}_{n = 0}^\infty \).     

Approximation of entire functions of exponential type by exponential sums<Superscript>*</Superscript>

Let K be a compact set in \( {{\mathbb R}^n} \). For \( 1 \leqslant p \leqslant \infty \), the Bernstein space \( B_K^p \) is the Banach space of all functions \( f \in {L^p}\left( {{{\mathbb R}^n}} \right) \)such that their Fourier transform in a distributional sense is supported on K. If \( f \in B_K^p \), then f is continuous on \( {{\mathbb R}^n} \) and has an extension onto the complex space \( {{\mathbb C}^n} \) to an entire function of exponential type K. We study the approximation of functions in \( B_K^p \) by finite τ -periodic exponential sums of the form

$ \sum\limits_m {{c_m}{e^{2\pi {\text{i}}\left( {x,m} \right)/\tau }}} $

in the \( {L^p}\left( {\tau {{\left[ { - 1/2,1/2} \right]}^n}} \right) \)-norm as τ → ∞ when K is a polytope in \( {{\mathbb R}^n} \).

     

The Regularity Problems with Data in Hardy–Sobolev Spaces for Singular Schrödinger Equation in Lipschitz Domains

Let the nonnegative singular potential V belong to the reverse Hölder class \({\mathcal B}_n\) on \({\mathbb R}^n\), and let (n???1)/n?p?≤?2, we establish the solvability and derivative estimates for the solutions to the Neumann problem and the regularity problem of the Schrödinger equation ??Δu?+?Vu?=?0 in a connected Lipschitz domain Ω, with boundary data in the Hardy space \(H^p(\partial \Omega)\) and the modified Hardy–Sobolev space \(H_{1, V}^p(\partial \Omega)\) related to the potential V. To deal with the H ^p regularity problem, we construct a new characterization of the atomic decomposition for \(H_{1, V}^p(\partial \Omega)\) space. The invertibility of the boundary layer potentials on Hardy spaces and Hölder spaces are shown in this paper.     

Strong extensions for q-summing o perators acting in p-convex Banach function s paces for $$1 \le p \le q$$1≤ p≤ q

Let \(1\le p\le q<\infty \) and let X be a p-convex Banach function space over a \(\sigma \)-finite measure \(\mu \). We combine the structure of the spaces \(L^p(\mu )\) and \(L^q(\xi )\) for constructing the new space \(S_{X_p}^{\,q}(\xi )\), where \(\xi \) is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every q-summing operator T defined on X can be continuously (strongly) extended to \(S_{X_p}^{\,q}(\xi )\). Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-summing operators through \(L^q\)-spaces when \(1 \le q \le p\). Thus, our result completes the picture, showing what happens in the complementary case \(1\le p\le q\).     

Generation of Semigroups for Vector-Valued Pseudodifferential Operators on the Torus

We consider toroidal pseudodifferential operators with operator-valued symbols, their mapping properties and the generation of analytic semigroups on vector-valued Besov and Sobolev spaces. Here, we restrict ourselves to pseudodifferential operators with x-independent symbols (Fourier multipliers). We show that a parabolic toroidal pseudodifferential operator generates an analytic semigroup on the Besov space \(B_{pq}^s({\mathbb T}^n,E)\) and on the Sobolev space \(W_p^k({\mathbb T}^n,E)\), where E is an arbitrary Banach space, \(1\le p,q\le \infty \), \(s\in {\mathbb R}\) and \(k\in {\mathbb N}_0\). For the proof of the Sobolev space result, we establish a uniform estimate on the kernel which is given as an infinite parameter-dependent sum. An application to abstract non-autonomous periodic pseudodifferential Cauchy problems gives the existence and uniqueness of classical solutions for such problems.     

One-sided invertibility of discrete operators and their applications

For \(p\in [1,\infty ]\), we establish criteria for the one-sided invertibility of binomial discrete difference operators \({{\mathcal {A}}}=aI-bV\) on the space \(l^p=l^p(\mathbb {Z})\), where \(a,b\in l^\infty \), I is the identity operator and the isometric shift operator V is given on functions \(f\in l^p\) by \((Vf)(n)=f(n+1)\) for all \(n\in \mathbb {Z}\). Applying these criteria, we obtain criteria for the one-sided invertibility of binomial functional operators \(A=aI-bU_\alpha \) on the Lebesgue space \(L^p(\mathbb {R}_+)\) for every \(p\in [1,\infty ]\), where \(a,b\in L^\infty (\mathbb {R}_+)\), \(\alpha \) is an orientation-preserving bi-Lipschitz homeomorphism of \([0,+\infty ]\) onto itself with only two fixed points 0 and \(\infty \), and \(U_\alpha \) is the isometric weighted shift operator on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f= (\alpha ^\prime )^{1/p}(f\circ \alpha )\). Applications of binomial discrete operators to interpolation theory are given.     

On the singular series for primes in arithmetic progressions<Superscript>*</Superscript>

Using \( \mathfrak{G} \)(d, k) to denote the singular series for primes in arithmetic progressions and using the word “tail” to denote the difference of \( \mathfrak{G} \)(d, k) and its partial sums, we establish some asymptotic formulas for weighted sums of the square of the tail and so give more information on the singular series \( \mathfrak{G} \)(d, k).     

Operator maps of Jensen-type

Let \(\mathbb {B}_J({\mathcal {H}})\) denote the set of self-adjoint operators acting on a Hilbert space \(\mathcal {H}\) with spectra contained in an open interval J. A map \(\Phi :\mathbb {B}_J({\mathcal {H}})\rightarrow {{\mathbb {B}}}({\mathcal {H}})_\text {sa} \) is said to be of Jensen-type if

$$\begin{aligned} \Phi (C^*AC+D^*BD)\le C^*\Phi (A)C+D^*\Phi (B)D \end{aligned}$$

for all \( A, B \in \mathbb {B}_J({\mathcal {H}})\) and bounded linear operators C, D acting on \( \mathcal {H} \) with \( C^*C+D^*D=I\), where I denotes the identity operator. We show that a Jensen-type map on an infinite dimensional Hilbert space is of the form \(\Phi (A)=f(A)\) for some operator convex function f defined in J.

     

Analytic sets and extension of holomorphic maps of positive codimension

Let D, \(D'\) be arbitrary domains in \({\mathbb C}^n\) and \({\mathbb C}^N\) respectively, \(1<n\le N\), both possibly unbounded and \(M \subseteq \partial D\), \(M'\subseteq \partial D'\) be open pieces of the boundaries. Suppose that \(\partial D\) is smooth real-analytic and minimal in an open neighborhood of \({\bar{M}}\) and \(\partial D'\) is smooth real-algebraic and minimal in an open neighborhood of \({\bar{M}'}\). Let \(f: D\rightarrow D'\) be a holomorphic mapping such that the cluster set \(\mathrm{cl}_{f}(M)\) does not intersect \(D'\). It is proved that if the cluster set \(\mathrm{cl}_{f}(p)\) of some point \(p\in M\) contains some point \(q\in M'\) and the graph of f extends as an analytic set to a neighborhood of \((p, q)\in {\mathbb {C}}^n \times {\mathbb C}^N\), then f extends as a holomorphic map to a dense subset of some neighborhood of p. If in addition, \(M =\partial D\), \(M'=\partial D'\) and \(M'\) is compact, then f extends holomorphically across an open dense subset of \(\partial D\).     

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R~n×R~m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R~n× R~m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R~n× R~m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R~n× R~m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R~n× R~m) to L~φ(R~n× R~m)and from H~φ_A(R~n×R~m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R~n× R~m and are new even for classical product Orlicz-Hardy spaces.     

Generating Hyperbolic Singularities in Semitoric Systems Via Hopf Bifurcations

Let \((M,\Omega )\) be a connected symplectic 4-manifold and let \(F=(J,H) :M\rightarrow \mathbb {R}^2\) be a completely integrable system on M with only non-degenerate singularities. Assume that F does not have singularities with hyperbolic blocks and that \(p_1,\ldots ,p_n\) are the focus–focus singularities of F. For each subset \(S=\{i_1,\ldots ,i_j\}\), we will show how to modify F locally around any \(p_i, i \in S\), in order to create a new integrable system \(\widetilde{F}=(J, \widetilde{H}) :M \rightarrow \mathbb {R}^2\) such that its classical spectrum \(\widetilde{F}(M)\) contains j smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of \(\widetilde{F}\). Moreover the focus–focus singularities of \(\widetilde{F}\) are precisely \(p_i\), \(i \in \{1,\ldots ,n\} \setminus S\). The proof is based on Eliasson’s linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.     

Symmetric Pairs and Self-Adjoint Extensions of Operators,with Applications to Energy Networks

We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator A on a Hilbert space \(\mathcal {H}\), by means of a symmetric pair of operators. A symmetric pair is comprised of densely defined operators \(J: \mathcal {H}_1 \rightarrow \mathcal {H}_2\) and \(K: \mathcal {H}_2 \rightarrow \mathcal {H}_1\) which are compatible in a certain sense. With the appropriate definitions of \(\mathcal {H}_1\) and J in terms of A and \(\mathcal {H}\), we show that \((\textit{JJ}^\star )^{-1}\) is the Friedrichs extension of A. Furthermore, we use related ideas (including the notion of unbounded containment) to construct a generalization of the construction of the Krein extension of A as laid out in a previous paper of the authors. These results are applied to the study of the graph Laplacian on infinite networks, in relation to the Hilbert spaces \(\ell ^2(G)\) and \(\mathcal {H}_{\mathcal {E}}\) (the energy space).     

Algebraic conditions for additive functions over the reals and over finite fields

Let C be an affine plane curve. We consider additive functions \(f{:}\; K\rightarrow K\) for which \(f(x)f(y)=0\), whenever \((x,y)\in C\). We show that if \(K=\mathbb {R}\) and C is the hyperbola with defining equation \(xy=1\), then there exist nonzero additive functions with this property. Moreover, we show that such a nonzero f exists for a field K if and only if K is transcendental over \(\mathbb Q\) or over \(\mathbb {F}_p\), the finite field with p elements. We also consider the general question when K is a finite field. We show that if the degree of the curve C is large enough compared to the characteristic of K, then f must be identically zero.