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 \(\partial D\).     

Fully robust one-sided cross-validation for regression functions

Fully robust OSCV is a modification of the OSCV method that produces consistent bandwidths in the cases of smooth and nonsmooth regression functions. We propose the practical implementation of the method based on the robust cross-validation kernel \(H_I\) in the case when the Gaussian kernel \(\phi \) is used in computing the resulting regression estimate. The kernel \(H_I\) produces practically unbiased bandwidths in the smooth and nonsmooth cases and performs adequately in the data examples. Negative tails of \(H_I\) occasionally result in unacceptably wiggly OSCV curves in the neighborhood of zero. This problem can be resolved by selecting the bandwidth from the largest local minimum of the curve. Further search for the robust kernels with desired properties brought us to consider the quartic kernel for the cross-validation purposes. The quartic kernel is almost robust in the sense that in the nonsmooth case it substantially reduces the asymptotic relative bandwidth bias compared to \(\phi \). However, the quartic kernel is found to produce more variable bandwidths compared to \(\phi \). Nevertheless, the quartic kernel has an advantage of producing smoother OSCV curves compared to \(H_I\). A simplified scale-free version of the OSCV method based on a rescaled one-sided kernel is proposed.     

Symmetry properties of resolving sets and metric bases in hypercubes

In this paper we consider some special characteristics of distances between vertices in the \(n\)-dimensional hypercube graph \(Q_n\) and, as a consequence, the corresponding symmetry properties of its resolving sets. It is illustrated how these properties can be implemented within a simple greedy heuristic in order to find efficiently an upper bound of the so called metric dimension \(\beta (Q_n)\) of \(Q_n\), i.e. the minimal cardinality of a resolving set in \(Q_n\). This heuristic was applied to generate upper bounds of \(\beta (Q_n)\) for \(n\) up to \(22\), which are for \(n\ge 19\) better than the existing ones. Starting from these new bounds, some existing upper bounds for \(23\le n\le 90\) are improved by a dynamic programming procedure.     

Fourier Spectrum Characterizations of $$H^{p}$$ Spaces on Tubes Over Cones for $$1\le p \le \infty $$1≤p≤∞

We give Fourier spectrum characterizations of functions in the Hardy \(H^p\) spaces on tubes for \(1\le p \le \infty .\) For \(F\in L^p(\mathbb {R}^n), \) we show that F is the non-tangential boundary limit of a function in a Hardy space, \(H^{p}(T_\Gamma ),\) where \(\Gamma \) is an open cone of \(\mathbb {R}^n\) and \(T_\Gamma \) is the related tube in \(\mathbb {C}^n,\) if and only if the classical or the distributional Fourier transform of F is supported in \(\Gamma ^*,\) where \(\Gamma ^*\) is the dual cone of \(\Gamma .\) This generalizes the results of Stein and Weiss for \(p=2\) in the same context, as well as those of Qian et al. in one complex variable for \(1\le p\le \infty .\) Furthermore, we extend the Poisson and Cauchy integral representation formulas to the \(H^p\) spaces on tubes for \(p\in [1, \infty ]\) and \(p\in [1,\infty ),\) with, respectively, the two types of representations.     

Derivations Vanishing Identities Involving Generalized Derivations and Multilinear Polynomial in Prime Rings

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, \(f(x_1,\ldots ,x_n)\) be a multilinear polynomial over C, which is not central valued on R. Suppose that d is a non-zero derivation of R, F and G are two generalized derivations of R such that \(d\{F(u)u-uG^2(u)\}=0\) for all \(u\in f(R)\). Then one of the following holds:

1. (i)
   there exist \(a, b, p\in U\), \(\lambda \in C\) such that \(F(x)=\lambda x+bx+xa^2\), \(G(x)=ax\), \(d(x)=[p, x]\) for all \(x\in R\) with \([p, b]=0\) and \(f(x_1,\ldots , x_n)^2\) is central valued on R;
    
2. (ii)
   there exist \(a, b, p\in U\) such that \(F(x)=ax\), \(G(x)=xb\), \(d(x)=[p,x]\) for all \(x\in R\) and \(f(x_1,\ldots , x_n)^2\) is central valued on R with \([p, a-b^2]=0\);
    
3. (iii)
   there exist \(a\in U\) such that \(F(x)=xa^2\) and \(G(x)=ax\) for all \(x\in R\);
    
4. (iv)
   there exists \(a\in U\) such that \(F(x)=a^2x\) and \(G(x)=xa\) for all \(x\in R\) with \(a^2\in C\);
    
5. (v)
   there exist \(a, p\in U\), \(\lambda , \alpha , \mu \in C\) such that \(F(x)=\lambda x-a^2x\), \(G(x)=xa\) and \(d(x)=[p,x]\) for all \(x\in R\) with \(a^2=\mu -\alpha p\) and \(\alpha p^2+(\lambda -2\mu ) p\in C\);
    
6. (vi)
   there exist \(a\in U\), \(\lambda \in C\) such that R satisfies \(s_4\) and either \(F(x)=\lambda x+xa^2\), \(G(x)=ax\) or \(F(x)=\lambda x-a^2x\), \(G(x)=xa\) for all \(x\in R\).
    

     

Nodal surfaces with obstructed deformations

In this text we show that the deformation space of a nodal surface X of degree d is smooth and of the expected dimension if \(d\le 7\) or \(d\ge 8\) and X has at most \(4d-5\) nodes (The case \(d\le 7\) was previously covered by Alexandru Dimca using different techniques). For \(d\ge 8\) we give explicit examples of nodal surfaces with \(4d-4\) nodes, for which the tangent space to the deformation space has larger dimension than expected.     

Endotrivial modules for the general linear group in a nondefining characteristic

Suppose that \(G\) is a finite group such that \(\mathrm{SL }(n,q)\subseteq G \subseteq \mathrm{GL }(n,q)\), and that \(Z\) is a central subgroup of \(G\). Let \(T(G/Z)\) be the abelian group of equivalence classes of endotrivial \(k(G/Z)\)-modules, where \(k\) is an algebraically closed field of characteristic \(p\) not dividing \(q\). We show that the torsion free rank of \(T(G/Z)\) is at most one, and we determine \(T(G/Z)\) in the case that the Sylow \(p\)-subgroup of \(G\) is abelian and nontrivial. The proofs for the torsion subgroup of \(T(G/Z)\) use the theory of Young modules for \(\mathrm{GL }(n,q)\) and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules.     

Classifying First Extremal Points for a Fractional Boundary Value Problem with a Fractional Boundary Condition

Let \(n\in \mathbb {N}\), \(n\ge 2\), \(\beta >0\) fixed, and \(0<b\le \beta \). For \(n-1<\alpha \le n\), we look to classify extremal points for the fractional differential equation \(D_{0^+}^{\alpha }u+p(t) u=0\), satisfying the boundary conditions \(u^{(i)}(0)=0\), \(i=0,\ldots ,n-2\), \(D_{0^+}^\gamma u(b)=0\), where p(t) is a continuous nonnegative function on \([0,\beta ]\) which does not vanish identically on any nondegenerate compact subinterval of \([0,\beta ]\). Using the theory of Krein and Rutman, first extremal points of this boundary value problem are classified. As an application, the results are applied, along with a fixed-point theorem, to show the existence of a solution of a nonlinear fractional boundary value problem.     

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 \(p\ge 1\).     

Ramanujan-type congruences for overpartitions modulo 16

Let \(\bar{p}(n)\) denote the number of overpartitions of \(n\). Recently, Fortin–Jacob–Mathieu and Hirschhorn–Sellers independently obtained 2-, 3- and 4-dissections of the generating function for \(\bar{p}(n)\) and derived a number of congruences for \(\bar{p}(n)\) modulo 4, 8 and 64 including \(\bar{p}(8n+7)\equiv 0 \pmod {64}\) for \(n\ge 0\). In this paper, we give a 16-dissection of the generating function for \(\bar{p}(n)\) modulo 16 and show that \(\bar{p}(16n+14)\equiv 0\pmod {16}\) for \(n\ge 0\). Moreover, using the \(2\)-adic expansion of the generating function for \(\bar{p}(n)\) according to Mahlburg, we obtain that \(\bar{p}(\ell ^2n+r\ell )\equiv 0\pmod {16}\), where \(n\ge 0\), \(\ell \equiv -1\pmod {8}\) is an odd prime and \(r\) is a positive integer with \(\ell \not \mid r\). In particular, for \(\ell =7\) and \(n\ge 0\), we get \(\bar{p}(49n+7)\equiv 0\pmod {16}\) and \(\bar{p}(49n+14)\equiv 0\pmod {16}\). We also find four congruence relations: \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n) \pmod {16}\) for \(n\ge 0\), \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {32}\) where \(n\) is not a square of an odd positive integer, \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {64}\) for \(n\not \equiv 1,2,5\pmod {8}\) and \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {128}\) for \(n\equiv 0\pmod {4}\).     

Bergman Subspaces and Subkernels: Degenerate $$L^p$$ Mapping and Zeroes

Regularity and irregularity of the Bergman projection on \(L^p\) spaces is established on a natural family of bounded, pseudoconvex domains. The family is parameterized by a real variable \(\gamma \). A surprising consequence of the analysis is that, whenever \(\gamma \) is irrational, the Bergman projection is bounded only for \(p=2\).     

The $$$$-Affine Caacity

In this paper, the \(p\)-affine capacity is introduced for \(1<p<n\) and then developed to discover the upper and lower isocapacitary inequalities that strengthen optimally both the Maz’ya \(p\)-isocapacitary inequality and the Lutwak–Yang–Zhang \(L_p\) affine isoperimetric inequality over the \(n\)-dimensional Euclidean space \({\mathbb {R}}^{n}\).     

On homogeneous hypersurfaces in $${\mathbb {C}}^3$$

We consider a family \(M_t^n\), with \(n\geqslant 2\), \(t>1\), of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in  \({\mathbb {C}}^n\) due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of \(M_t^n\) in  \({\mathbb {C}}^n\) for \(n=3,7\). In our earlier article we showed that \(M_t^7\) is not embeddable in  \({\mathbb {C}}^7\) for every t and that \(M_t^3\) is embeddable in  \({\mathbb {C}}^3\) for all \(1<t<1+10^{-6}\). In the present paper, we improve on the latter result by showing that the embeddability of \(M_t^3\) in fact takes place for \(1<t<\sqrt{(2+\sqrt{2})/3}\). This is achieved by analyzing the explicit totally real embedding of the sphere \(S^3\) in \({\mathbb {C}}^3\) constructed by Ahern and Rudin. For \(t\geqslant {\sqrt{(2+\sqrt{2})/3}}\), the problem of the embeddability of \(M_t^3\) remains open.     

Existence of self-Cheeger sets on Riemannian manifolds

Let \(({\mathcal M},g)\) be a smooth compact Riemannian manifold of dimension \(N\ge 2\). We prove the existence of a family \((\Omega _\varepsilon )_{\varepsilon \in (0,\varepsilon _0)}\) of self-Cheeger sets in \(({\mathcal M},g)\). The domains \(\Omega _\varepsilon \subset {\mathcal M}\) are perturbations of geodesic balls of radius \(\varepsilon \) centered at \(p \in {\mathcal M}\), and in particular, if \(p_0\) is a non-degenerate critical point of the scalar curvature of g, then the family \((\partial \Omega _\varepsilon )_{\varepsilon \in (0,\varepsilon _0)}\) constitutes a smooth foliation of a neighborhood of \(p_0\).     

Polynomial convergence of two higher order interior-point methods for $$P_*(\kappa )$$-LCP in a wide neighborhood of the central path

In this paper, we propose two interior-point methods for solving \(P_*(\kappa )\)-linear complementarity problems (\(P_*(\kappa )\)-LCPs): a high order large update path following method and a high order corrector–predictor method. Both algorithms generate sequences of iterates in the wide neighborhood \((\mathcal {N}_{2,\tau }^-(\alpha ))\) of the central path introduced by Ai and Zhang. The methods do not depend on the handicap \(\kappa \) of the problem so that they work for any \(P_*(\kappa )\)-LCP . They have \(O((1 +\kappa )\sqrt{n}L)\) iteration complexity, the best-known iteration complexity obtained so far by any interior-point method for solving \(P_*(\kappa )\)-LCP. The high order corrector–predictor algorithm is superlinearly convergent with Q-order \((m_p+1)\) for problems that admit a strict complementarity solution and \((m_p+1)/2\) for general problems, where \(m_p\) is the order of the predictor step.     

An $$H^1$$-Galerkin mixed finite element method for time fractional reaction–diffusion equation

In this article, an \(H^1\)-Galerkin mixed finite element (MFE) method for solving time fractional reaction–diffusion equation is presented. The optimal time convergence order \(O(\varDelta t^{2-\alpha })\) and the optimal spatial rate of convergence in \(H^1\) and \(L^2\)-norms for variable \(u\) and its gradient \(\sigma \) are derived. Moreover, some numerical results are shown to support our theoretical analysis.     

Elimination of extremal index zeroes from generic paths of closed $$$$-forms

Let \( \alpha \) be a Morse closed \( 1 \)-form of a smooth \( n \)-dimensional manifold \( M \). The zeroes of \( \alpha \) of index \( 0 \) or \( n \) are called centers. It is known that every non-vanishing de Rham cohomology class \( u \) contains a Morse representative without centers. The result of this paper is the one-parameter analogue of the last statement: every generic path \( (\alpha _t)_{ t\in [0,1] }\) of closed \( 1 \)-forms in a fixed class \( u\ne 0 \) such that \( \alpha _0,\alpha _1 \) have no centers, can be modified relatively to its extremities to another such path \( (\beta _t)_{t \in [0,1]} \) having no center at all.     

Sharp Bounds for the Ratio of Modified Bessel Functions

Let \(I_{\nu }( x) \) be the modified Bessel functions of the first kind of order \(\nu \), and \(S_{p,\nu }( x) =W_{\nu }( x) ^{2}-2pW_{\nu }( x) -x^{2}\) with \(W_{\nu }( x) =xI_{\nu }( x) /I_{\nu +1}( x) \). We achieve necessary and sufficient conditions for the inequality \(S_{p,\nu }( x) <u\) or \(S_{p,\nu }( x) >l\) to hold for \(x>0\) by establishing the monotonicity of \(S_{p,\nu }(x)\) in \(x\in ( 0,\infty ) \) with \(\nu >-3/2\). In addition, the best parameters p and q are obtained to the inequality \(W_{\nu }( x) <( >) p+\sqrt{ x^{2}+q^{2}}\) for \(x>0\). Our main achievements improve some known results, and it seems to answer an open problem recently posed by Hornik and Grün (J Math Anal Appl 408:91–101, 2013).     

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 First Passage Time of a Stable Process Conditioned to Not Overshoot

Consider a stable Lévy process \(X=(X_t,t\ge 0)\) and let \(T_{x}\), for \(x>0\), denote the first passage time of \(X\) above the level \(x\). In this work, we give an alternative proof of the absolute continuity of the law of \(T_{x}\) and we obtain a new expression for its density function. Our constructive approach provides a new insight into the study of the law of \(T_{x}\). The random variable \(T_{x}^{0}\), defined as the limit of \(T_{x}\) when the corresponding overshoot tends to \(0\), plays an important role in obtaining these results. Moreover, we establish a relation between the random variable \(T_{x}^{0}\) and the dual process conditioned to die at \(0\). This relation allows us to link the expression of the density function of the law of \(T_{x}\) presented in this paper to the already known results on this topic.