The graph of minimal distances of bent functions and its properties

A notion of the graph of minimal distances of bent functions is introduced. It is an undirected graph (V, E) where V is the set of all bent functions in 2k variables and \((f, g) \in E\) if the Hamming distance between f and g is equal to \(2^k\). It is shown that the maximum degree of the graph is equal to \(2^k (2^1 + 1) (2^2 + 1) \cdots (2^k + 1)\) and all its vertices of maximum degree are quadratic bent functions. It is obtained that the degree of a vertex from Maiorana—McFarland class is not less than \(2^{2k + 1} - 2^k\). It is proven that the graph is connected for \(2k = 2, 4, 6\), disconnected for \(2k \ge 10\) and its subgraph induced by all functions EA-equivalent to Maiorana—McFarland bent functions is connected.     

Diagonalization of the Finite Hilbert Transform on Two Adjacent Intervals

We continue the study of stability of solving the interior problem of tomography. The starting point is the Gelfand–Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function f along a collection of lines. Pick one such line, call it the x-axis, and assume that the function to be reconstructed depends on a one-dimensional argument by restricting f to the x-axis. Let \(I_1\) be the interval where f is supported, and \(I_2\) be the interval where the Hilbert transform of f can be computed using the Gelfand–Graev formula. The equation to be solved is \(\left. {\mathcal {H}}_1 f=g\right| _{I_2}\), where \({\mathcal {H}}_1\) is the FHT that integrates over \(I_1\) and gives the result on \(I_2\), i.e. \({\mathcal {H}}_1: L^2(I_1)\rightarrow L^2(I_2)\). In the case of complete data, \(I_1\subset I_2\), and the classical FHT inversion formula reconstructs f in a stable fashion. In the case of interior problem (i.e., when the tomographic data are truncated), \(I_1\) is no longer a subset of \(I_2\), and the inversion problems becomes severely unstable. By using a differential operator L that commutes with \({\mathcal {H}}_1\), one can obtain the singular value decomposition of \({\mathcal {H}}_1\). Then the rate of decay of singular values of \({\mathcal {H}}_1\) is the measure of instability of finding f. Depending on the available tomographic data, different relative positions of the intervals \(I_{1,2}\) are possible. The cases when \(I_1\) and \(I_2\) are at a positive distance from each other or when they overlap have been investigated already. It was shown that in both cases the spectrum of the operator \({\mathcal {H}}_1^*{\mathcal {H}}_1\) is discrete, and the asymptotics of its eigenvalues \(\sigma _n\) as \(n\rightarrow \infty \) has been obtained. In this paper we consider the case when the intervals \(I_1=(a_1,0)\) and \(I_2=(0,a_2)\) are adjacent. Here \(a_1 < 0 < a_2\). Using recent developments in the Titchmarsh–Weyl theory, we show that the operator L corresponding to two touching intervals has only continuous spectrum and obtain two isometric transformations \(U_1\), \(U_2\), such that \(U_2{\mathcal {H}}_1 U_1^*\) is the multiplication operator with the function \(\sigma (\lambda )\), \(\lambda \ge (a_1^2+a_2^2)/8\). Here \(\lambda \) is the spectral parameter. Then we show that \(\sigma (\lambda )\rightarrow 0\) as \(\lambda \rightarrow \infty \) exponentially fast. This implies that the problem of finding f is severely ill-posed. We also obtain the leading asymptotic behavior of the kernels involved in the integral operators \(U_1\), \(U_2\) as \(\lambda \rightarrow \infty \). When the intervals are symmetric, i.e. \(-a_1=a_2\), the operators \(U_1\), \(U_2\) are obtained explicitly in terms of hypergeometric functions.     

Generalized Fibonacci numbers of the form $$wx^{2}+1$$

Let \(P\ge 3\) be an integer and let \((U_{n})\) and \((V_{n})\) denote generalized Fibonacci and Lucas sequences defined by \(U_{0}=0,U_{1}=1\); \( V_{0}=2,V_{1}=P,\) and \(U_{n+1}=PU_{n}-U_{n-1}\), \(V_{n+1}=PV_{n}-V_{n-1}\) for \(n\ge 1.\) In this study, when P is odd, we solve the equation \( U_{n}=wx^{2}+1\) for \(w=1,2,3,5,6,7,10.\) After then, we solve some Diophantine equations utilizing solutions of these equations.     

Averaged Wave Operators and Complex-symmetric Operators

We study the behaviour of sequences \(U_2^n X U_1^{-n}\), where \(U_1, U_2\) are unitary operators, whose spectral measures are singular with respect to the Lebesgue measure, and the commutator \(XU_1-U_2X\) is small in a sense. The conjecture about the weak averaged convergence of the difference \(U_2^n X U_1^{-n}-U_2^{-n} X U_1^n\) to the zero operator is discussed and its connection with complex-symmetric operators is established in a general situation. For a model case where \(U_1=U_2\) is the unitary operator of multiplication by z on \(L^2(\mu )\), sufficient conditions for the convergence as in the Conjecture are given in terms of kernels of integral operators.     

Closures of Hardy and Hardy–Sobolev Spaces in the Bloch Type Space on the Unit Ball

For \(0<\alpha <\infty \), \(0<p<\infty \) and \(0<s<\infty \), we characterize the closures in the \(\alpha \)-Bloch norm of \(\alpha \)-Bloch functions that are in a Hardy space \(H^p\) and in a Hardy–Sobolev space \(H^p_s\) on the unit ball of \(\mathbb {C}^n\).     

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.     

Meromorphic tensor equivalence for Yangians and quantum loop algebras

Let \(\mathfrak{g}\) be a complex semisimple Lie algebra, and \(Y_{\hbar }(\mathfrak{g})\), \(U_{q}(L\mathfrak{g})\) the corresponding Yangian and quantum loop algebra, with deformation parameters related by \(q=e^{\pi \iota \hbar }\). When \(\hbar \) is not a rational number, we constructed in Gautam and Toledano Laredo (J. Am. Math. Soc. 29:775, 2016) a faithful functor \(\Gamma \) from the category of finite-dimensional representations of \(Y_{\hbar }(\mathfrak{g})\) to those of \(U_{q}(L \mathfrak{g})\). The functor \(\Gamma \) is governed by the additive difference equations defined by the commuting fields of the Yangian, and restricts to an equivalence on a subcategory of \(\operatorname{Rep}_{\operatorname{fd}}(Y_{\hbar }(\mathfrak{g}))\) defined by choosing a branch of the logarithm. In this paper, we construct a tensor structure on \(\Gamma \) and show that, if \(|q|\neq 1\), it yields an equivalence of meromorphic braided tensor categories, when \(Y_{\hbar }(\mathfrak{g})\) and \(U_{q}(L\mathfrak{g})\) are endowed with the deformed Drinfeld coproducts and the commutative part of their universal \(R\)-matrices. This proves in particular the Kohno–Drinfeld theorem for the abelian \(q\)KZ equations defined by \(Y_{\hbar }(\mathfrak{g})\). The tensor structure arises from the abelian \(q\)KZ equations defined by an appropriate regularisation of the commutative part of the \(R\)-matrix of \(Y_{\hbar }(\mathfrak{g})\).     

Reducibility for a Class of Analytic Multipliers on the Dirichlet Space

It is proved that if \(\phi \) is a finite Blaschke product with four zeros, then \(M_\phi \) is reducible on the Dirichlet space with norm \(\Vert \ \Vert \) if and only if \(\phi =\phi _1\circ \phi _2\), where \(\phi _1, \phi _2\) are Blaschke products and \(\phi _2\) is equivalent to \(z^2\). Also, the same reducibility of \(M_\phi \) with finite Blaschke product \(\phi \) on the Dirichlet space under the equivalent norms \(\Vert \ \Vert _1\) and \(\Vert \ \Vert _0\) is given.     

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

     

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.     

The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators

We investigate the weighted bounds for multilinear maximal functions and Calderón–Zygmund operators from \(L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\) to \(L^{p}(v_{\vec {w}})\), where \(1<p_1,\cdots ,p_m<\infty \) with \(1/{p_1}+\cdots +1/{p_m}=1/p\) and \(\vec {w}\) is a multiple \(A_{\vec {P}}\) weight. We prove the sharp bound for the multilinear maximal function for all such \(p_1,\ldots , p_m\) and prove the sharp bound for \(m\)-linear Calderón–Zymund operators when \(p\ge 1\).     

Some identities for Appell–Lerch sums and a universal mock theta function

In their paper “A survey of classical mock theta functions”, Gordon and McIntosh observed that the classical mock \(\theta \)-functions, including those found by Ramanujan, can be expressed in terms of two ‘universal’ mock \(\theta \)-functions denoted by \(g_{_{2}}\) and \(g_{_{3}}\). These functions are normalized level 2 and level 3 Appell–Lerch functions. In the survey paper, the authors list several identities for certain Appell–Lerch functions and refer the proofs to a future paper with this title, listed in their references as [GM3]. The purpose of this paper is to prove these identities. One of the identities removes the \( \theta \) -quotients from Kang’s formulas, which express \(g_{_{2}}\) and \({g}_{{_{3}}}\) in terms of Zwegers’ \(\mu \)-function and \( \theta \)-quotients.     

Homogeneous $$$$-difference euations and generating functions for $$$$-hypergeometric polynomials

In this paper we show how to deduce several types of generating functions for \(q\)-hypergeometric polynomials by the method of homogeneous \(q\)-difference equations. In addition, we build relations between transformation formulas and homogeneous \(q\)-difference equations. Moreover, we generalize the Andrews–Askey integral from the perspective of \(q\)-integrals by the method of homogeneous \(q\)-difference equations.     

On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts

The Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts \(\zeta (s+i\tau )\), \(\tau \in \mathbb {R}\), of the Riemann zeta-function. In the paper, we obtain a universality theorem on the approximation of analytic functions by discrete shifts \(\zeta (s+ix_kh)\), \(k\in \mathbb {N}\), \(h>0\), where \(\{x_k\}\subset \mathbb {R}\) is such that the sequence \(\{ax_k\}\) with every real \(a\ne 0\) is uniformly distributed modulo 1, \(1\le x_k\le k\) for all \(k\in \mathbb {N}\) and, for \(1\le k\), \(m\le N\), \(k\ne m\), the inequality \(|x_k-x_m| \ge y^{-1}_N\) holds with \(y_N> 0\) satisfying \(y_Nx_N\ll N\).     

Minimization of transformed $$L_1$$ penalty: theory,difference of convex function algorithm,and robust application in compressed sensing

We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed \(l_1\) (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates \(l_0\) and \(l_1\) norms through a nonnegative parameter \(a \in (0,+\infty )\), similar to \(l_p\) with \(p \in (0,1]\), and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem, and discuss the exact recovery of \(l_0\) norm minimal solution based on the null space property (NSP). We then prove the stable recovery of \(l_0\) norm minimal solution if the sensing matrix A satisfies a restricted isometry property (RIP). We formulated a normalized problem to overcome the lack of scaling property of the TL1 penalty function. For a general sensing matrix A, we show that the support set of a local minimizer corresponds to linearly independent columns of A. Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized constrained and unconstrained problems in CS. The DCATL1 algorithm involves outer and inner loops of iterations, one time matrix inversion, repeated shrinkage operations and matrix-vector multiplications. The inner loop concerns an \(l_1\) minimization problem on which we employ the Alternating Direction Method of Multipliers. For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value \(a=1\), and compare DCATL1 with other CS algorithms on two classes of sensing matrices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). Among existing algorithms, the iterated reweighted least squares method based on \(l_{1/2}\) norm is the best in sparse recovery for Gaussian matrices, and the DCA algorithm based on \(l_1\) minus \(l_2\) penalty is the best for over-sampled DCT matrices. We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with \(l_1\) minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix A. DCATL1 is also competitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.     

Gagliardo-Nirenberg inequalities for differential forms in Heisenberg groups

The \(L^1\)-Sobolev inequality states that for compactly supported functions u on the Euclidean n-space, the \(L^{n/(n-1)}\)-norm of a compactly supported function is controlled by the \(L^1\)-norm of its gradient. The generalization to differential forms (due to Lanzani and Stein and Bourgain and Brezis) is recent, and states that a the \(L^{n/(n-1)}\)-norm of a compactly supported differential h-form is controlled by the \(L^1\)-norm of its exterior differential du and its exterior codifferential \(\delta u\) (in special cases the \(L^1\)-norm must be replaced by the \(\mathcal H^1\)-Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms.     

Two Novel Characterizations of Self-Decomposability on the Half-Line

Two novel characterizations of self-decomposable Bernstein functions are provided. The first one is purely analytic, stating that a function \(\varPsi \) is the Bernstein function of a self-decomposable probability law \(\pi \) on the positive half-axis if and only if alternating sums of \(\varPsi \) satisfy certain monotonicity conditions. The second characterization is of probabilistic nature, showing that \(\varPsi \) is a self-decomposable Bernstein function if and only if a related d-variate function \(C_{\psi ,d}\), \(\psi :=\exp (-\varPsi )\), is a d-variate copula for each \(d \ge 2\). A canonical stochastic construction is presented, in which \(\pi \) (respectively \(\varPsi \)) determines the probability law of an exchangeable sequence of random variables \(\{U_k\}_{k\in {\mathbb {N}}}\) such that \((U_1,\ldots ,U_d) \sim C_{\psi ,d}\) for each \(d \ge 2\). The random variables \(\{U_k\}_{k\in {\mathbb {N}}},\) are i.i.d. conditioned on an increasing Sato process whose law is characterized by \(\varPsi \). The probability law of \(\{U_k\}_{k \in {\mathbb {N}}}\) is studied in quite some detail.     

Left annihilator of generalized derivations on Lie ideals in prime rings

Let \(R\) be a prime ring, \(L\) a noncentral Lie ideal of \(R\), \(F\) a generalized derivation with associated nonzero derivation \(d\) of \(R\). If \(a\in R\) such that \(a(d(u)^{l_1} F(u)^{l_2} d(u)^{l_3} F(u)^{l_4} \ldots F(u)^{l_k})^{n}=0\) for all \(u\in L\), where \(l_1,l_2,\ldots ,l_k\) are fixed non negative integers not all are zero and \(n\) is a fixed integer, then either \(a=0\) or \(R\) satisfies \(s_4\), the standard identity in four variables.     

Standard Jordan partitions with three parameters

We extend previous work on standard two-parameter Jordan partitions by Barry (Commun Algebra 43:4231–4246, 2015) to three parameters. Let \(J_r\) denote an \(r \times r\) matrix with minimal polynomial \((t-1)^r\) over a field F of characteristic p. For positive integers \(n_1\), \(n_2\), and \(n_3\) satisfying \(n_1 \le n_2 \le n_3\), the Jordan canonical form of the \(n_1 n_2 n_3 \times n_1 n_2 n_3\) matrix \(J_{n_1} \otimes J_{n_2} \otimes J_{n_3}\) has the form \(J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _m}\) where \(\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _m>0\) and \(\sum _{i=1}^m \lambda _i=n_1 n_2 n_3\). The partition \(\lambda (n_1,n_2,n_3:p)=(\lambda _1, \lambda _2,\ldots , \lambda _m)\) of \(n_1 n_2 n_3\), which depends on \(n_1\), \(n_2\), \(n_3\), and p, will be called a Jordan partition. We will define what we mean by a standard Jordan partition and give necessary and sufficient conditions for its existence.     

Marcinkiewicz-Type Spectral Multipliers on Hardy and Lebesgue Spaces on Product Spaces of Homogeneous Type

Let \(X_1\) and \(X_2\) be metric spaces equipped with doubling measures and let \(L_1\) and \(L_2\) be nonnegative self-adjoint operators acting on \(L^2(X_1)\) and \(L^2(X_2)\) respectively. We study multivariable spectral multipliers \(F(L_1, L_2)\) acting on the Cartesian product of \(X_1\) and \(X_2\). Under the assumptions of the finite propagation speed property and Plancherel or Stein–Tomas restriction type estimates on the operators \(L_1\) and \(L_2\), we show that if a function F satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator \(F(L_1, L_2)\) is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space \(X_1\times X_2\). We apply our results to the analysis of second-order elliptic operators in the product setting, specifically Riesz-transform-like operators and double Bochner–Riesz means.