Some theoretical comparisons of refined Ritz vectors and Ritz vectors

Refined projection methods proposed by the author have received attention internationally. We are concerned with a conventional projection method and its refined counterpart for computing approximations to a simple eigenpair (λ, ϰ) of a large matrix A. Given a subspace W that contains an approximation to ϰ, these two methods compute approximations (μ\(\tilde x\)) and (μ\(\hat x\)) to (λ, ϰ), respectively. We establish three results. First, the refined eigenvector approximation or simply the refined Ritz vector \(\hat x\) is unique as the deviation of ϰ from W approaches zero if λ is simple. Second, in terms of residual norm of the refined approximate eigenpair (μ, \(\hat x\)), we derive lower and upper bounds for the sine of the angle between the Ritz vector \(\tilde x\) and the refined eigenvector approximation \(\hat x\), and we prove that \(\tilde x \ne \hat x\) unless \(\hat x = x\). Third, we establish relationships between the residual norm \(\left\| {A\tilde x - \mu \tilde x} \right\|\) of the conventional methods and the residual norm \(\left\| {A\hat x - \mu \hat x} \right\|\) of the refined methods, and we show that the latter is always smaller than the former if (μ, \(\hat x\)) is not an exact eigenpair of A, indicating that the refined projection method is superior to the corresponding conventional counterpart.

     

Integral Van Vleck’s and Kannappan’s functional equations on semigroups

In this paper we study the solutions of the integral Van Vleck’s functional equation for the sine

$$\begin{aligned} \int _{S}f(x\tau (y)t)d\mu (t)-\int _{S}f(xyt)d\mu (t) =2f(x)f(y),\; x,y\in S \end{aligned}$$

and the integral Kannappan’s functional equation

$$\begin{aligned} \int _{S}f(xyt)d\mu (t)+\int _{S}f(x\tau (y)t)d\mu (t) =2f(x)f(y),\; x,y\in S, \end{aligned}$$

where S is a semigroup, \(\tau \) is an involution of S and \(\mu \) is a measure that is a linear combination of Dirac measures \((\delta _{z_{i}})_{i\in I}\), such that for all \(i\in I\), \(z_{i}\) is contained in the center of S. We express the solutions of the first equation by means of multiplicative functions on S, and we prove that the solutions of the second equation are closely related to the solutions of d’Alembert’s classic functional equation with involution.

     

Maximal Operators in the Spherical Mean Setting

In this paper, we prove an existence result for \(\mathcal {L}^{\infty }\)-solutions for a class of semilinear delay evolution inclusions with measures and subjected to nonlocal initial conditions of the form

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \mathrm{d}u(t)= \{Au(t)+f(t)\}\mathrm{d}t+\mathrm{d}h(t),&{}\quad t\in \mathbb {R}_+,\\ \displaystyle f(t)\in F(t,u_t),&{}\quad t\in \mathbb {R}_+,\\ \displaystyle u(t)=g(u)(t),&{}\quad t\in [\,-\tau ,0\,]. \end{array} \right. \end{aligned}$$

Here \(\tau \ge 0\), X is a Banach space, \(A:D(A)\subseteq X \rightarrow X \) is the infinitesimal generator of a \(C_0\)-semigroup, \(F:\mathbb {R}_+\times \mathcal {R}([\,-\tau ,0\,];X)\rightsquigarrow X\) is a u.s.c. multifunction with nonempty, convex and weakly compact values, \(h\in BV_{\mathrm{loc}}(\mathbb {R}_+;X)\) and the function \(g:\mathcal {R}_{b}(\mathbb {R}_+;X)\rightarrow \mathcal {R}([\,-\tau ,0\,];X)\) is nonexpansive.

     

Gaussian radial‐basis functions: Cardinal interpolation of ℓp and power‐growth data

Suppose λ is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal function \(L_\lambda (x) = \sum\nolimits_{k \in \mathbb{Z}} {c_k \exp ( - \lambda (x - k)^2 ),x \in \mathbb{R}} ,\) satisfying the interpolatory conditions \(L_\lambda (j) = \delta _{0j} ,j \in \mathbb{Z}.\) The paper considers the Gaussian cardinal interpolation operator

$(\mathcal{L}_\lambda {\text{y}})(x): = \sum\limits_{k \in \mathbb{Z}} {y_k L_\lambda (x - k),{\text{ y}} = (y_k )_{k \in \mathbb{Z}} ,{\text{ }}x \in \mathbb{R}} ,$

as a linear mapping from ℓ^p(ℤ) into L ^p(ℝ), 1≤ p ∞, and in particular, its behaviour as λ→0⁺. It is shown that \(\left\| {\mathcal{L}_\lambda } \right\|_p \) is uniformly bounded (in λ) for 1 < p < ∞, and that \(\left\| {\mathcal{L}_\lambda } \right\|_1 \asymp \log (1/\lambda )\) as λ→0⁺. The limiting behaviour is seen to be that of the classical Whittaker operator

$\mathcal{W}:{\text{y}} \mapsto \sum\limits_{k \in \mathbb{Z}} {y_k \frac{{\sin \pi (x - k)}}{{\pi (x - k)}}} ,$

in that \(\lim _{\lambda \to 0^ + } \left\| {\mathcal{L}_\lambda {\text{y}} - \mathcal{W}{\text{y}}} \right\|_p = 0,\) for every \({\text{y}} \in \ell ^p (\mathbb{Z}){\text{ and }}1 < p < \infty .\) It is further shown that the Gaussian cardinal interpolants to a function f which is the Fourier transform of a tempered distribution supported in (-π,π) converge locally uniformly to f as λ→0⁺. Multidimensional extensions of these results are also discussed.

     

Euclidean Distance Between Haar Orthogonal and Gaussian Matrices

In this work, we study a version of the general question of how well a Haar-distributed orthogonal matrix can be approximated by a random Gaussian matrix. Here, we consider a Gaussian random matrix \(Y_n\) of order n and apply to it the Gram–Schmidt orthonormalization procedure by columns to obtain a Haar-distributed orthogonal matrix \(U_n\). If \(F_i^m\) denotes the vector formed by the first m-coordinates of the ith row of \(Y_n-\sqrt{n}U_n\) and \(\alpha \,=\,\frac{m}{n}\), our main result shows that the Euclidean norm of \(F_i^m\) converges exponentially fast to \(\sqrt{ \big (2-\frac{4}{3} \frac{(1-(1 -\alpha )^{3/2})}{\alpha }\big )m}\), up to negligible terms. To show the extent of this result, we use it to study the convergence of the supremum norm \(\epsilon _n(m)\,=\,\sup _{1\le i \le n, 1\le j \le m} |y_{i,j}- \sqrt{n}u_{i,j}|\) and we find a coupling that improves by a factor \(\sqrt{2}\) the recently proved best known upper bound on \(\epsilon _n(m)\). Our main result also has applications in Quantum Information Theory.     

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

We consider the model space \(\mathbb {M}^{n}_{K}\) of constant curvature K and dimension \(n\ge 1\) (Euclidean space for \(K=0\), sphere for \(K>0\) and hyperbolic space for \(K<0\)), and we show that given a function \(\rho :[0,\infty )\rightarrow [0, \infty )\) with \(\rho (0)=\mathrm {dist}(x,y)\) there exists a coadapted coupling (X(t), Y(t)) of Brownian motions on \(\mathbb {M}^{n}_{K}\) starting at (x, y) such that \(\rho (t)=\mathrm {dist}(X(t),Y(t))\) for every \(t\ge 0\) if and only if \(\rho \) is continuous and satisfies for almost every \(t\ge 0\) the differential inequality

$$\begin{aligned} -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) \le \rho '(t)\le -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) +\tfrac{2(n-1)\sqrt{K}}{\sin (\sqrt{K}\rho (t))}. \end{aligned}$$

In other words, we characterize all coadapted couplings of Brownian motions on the model space \(\mathbb {M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho \) satisfying the above hypotheses.

     

$$\pi $$π-Formulas and Gray code

In previous papers we introduced a class of polynomials which follow the same recursive formula as the Lucas–Lehmer numbers, studying the distribution of their zeros and remarking that this distribution follows a sequence related to the binary Gray code. It allowed us to give an order for all the zeros of every polynomial \(L_n\). In this paper, the zeros, expressed in terms of nested radicals, are used to obtain two formulas for \(\pi \): the first can be seen as a generalization of the known formula

$$\begin{aligned} \pi =\lim _{n\rightarrow \infty } 2^{n+1}\cdot \sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{n}}, \end{aligned}$$

related to the smallest positive zero of \(L_n\); the second is an exact formula for \(\pi \) achieved thanks to some identities valid for \(L_n\).

     

A priori estimates for Donaldson’s equation over compact Hermitian manifolds

In this paper we prove a priori estimates for Donaldson equation’s $$\begin{aligned} \omega \wedge (\chi +\sqrt{-1}\partial \bar{\partial }\varphi )^{n-1} =e^{F}(\chi +\sqrt{-1}\partial \bar{\partial }\varphi )^{n}, \end{aligned}$$ over a compact complex manifold \(X\) of complex dimension \(n\) , where \(\omega \) and \(\chi \) are arbitrary Hermitian metrics. Our estimates answer a question of Tosatti-Weinkove (Asian J. Math. 14:19–40, 2010).     

A note on the algebra of operations for Hopf cohomology at odd primes

Let p be any prime, and let \({\mathcal B}(p)\) be the algebra of operations on the cohomology ring of any cocommutative \(\mathbb {F}_p\)-Hopf algebra. In this paper we show that when p is odd (and unlike the \(p=2\) case), \({\mathcal B}(p)\) cannot become an object in the Singer category of \(\mathbb {F}_p\)-algebras with coproducts, if we require that coproducts act on the generators of \({\mathcal B}(p)\) coherently with their nature of cohomology operations.

     

Scaling Limits for Mixed Kernels

Let \(\mu \) and \(\nu \) be measures supported on \(\left( -1,1\right) \) with corresponding orthonormal polynomials \(\left\{ p_{n}^{\mu }\right\} \) and \( \left\{ p_{n}^{\nu }\right\} \), respectively. Define the mixed kernel

$$\begin{aligned} K_{n}^{{\mu },\nu }\left( x,y\right) =\sum _{j=0}^{n-1}p_{j}^{\mu }\left( x\right) p_{j}^{\nu }\left( y\right) . \end{aligned}$$

We establish scaling limits such as

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{\pi \sqrt{1-\xi ^{2}}\sqrt{\mu ^{\prime }\left( \xi \right) \nu ^{\prime }\left( \xi \right) }}{n}K_{n}^{\mu ,\nu }\left( \xi +\frac{a\pi \sqrt{1-\xi ^{2}}}{n},\xi +\frac{b\pi \sqrt{1-\xi ^{2}}}{n}\right) \\&\quad =S\left( \frac{\pi \left( a-b\right) }{2}\right) \cos \left( \frac{\pi \left( a-b\right) }{2}+B\left( \xi \right) \right) , \end{aligned}$$

where \(S\left( t\right) =\frac{\sin t}{t}\) is the sinc kernel, and \(B\left( \xi \right) \) depends on \({\mu },\nu \) and \(\xi \). This reduces to the classical universality limit in the bulk when \(\mu =\nu \). We deduce applications to the zero distribution of \(K_{n}^{{\mu },\nu }\), and asymptotics for its derivatives.

     

Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\). We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in \(\mathbb {S}^3 \times \mathbb {S}^3\). We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\) can only be \(\frac{2}{\sqrt{3}}\) or \(\frac{4}{\sqrt{3}}\).     

On the periodicity of continued fractions in elliptic fields

Article [1] raised the question of the finiteness of the number of square-free polynomials f ∈ ?[h] of fixed degree for which \(\sqrt f \) has periodic continued fraction expansion in the field ?((h)) and the fields ?(h)(\(\sqrt f \)) are not isomorphic to one another and to fields of the form ?(h)\(\left( {\sqrt {c{h^n} + 1} } \right)\), where c ∈ ?* and n ∈ ?. In this paper, we give a positive answer to this question for an elliptic field ?(h)(\(\sqrt f \)) in the case deg f = 3.     

Sur un problème de capitulation du corps ℚ\sqrt {p_1 p_2 } , i dont le 2-groupe de classes est élémentaire

Let p ₁ ≡ p ₂ ≡ 1 (mod 8) be primes such that \(\left( {\tfrac{{p_1 }} {{p_2 }}} \right) = - 1\) and \(\left( {\tfrac{2} {{a + b}}} \right) = - 1\) , where p ₁ p ₂ = a ²+b ². Let \(i = \sqrt { - 1} \) , d = p ₁ p ₂, \(\Bbbk = \mathbb{Q}(\sqrt {d,} i),\Bbbk _2^{(1)} \) be the Hilbert 2-class field and \(\Bbbk ^{(*)} = \mathbb{Q}(\sqrt {p_1 } ,\sqrt {p_2 } ,i)\) be the genus field of \(\Bbbk \) . The 2-part \(C_{\Bbbk ,2} \) of the class group of \(\Bbbk \) is of type (2, 2, 2), so \(\Bbbk _2^{(1)} \) contains seven unramified quadratic extensions \(\mathbb{K}_j /\Bbbk \) and seven unramified biquadratic extensions \(\mathbb{L}_j /\Bbbk \) . Our goal is to determine the fourteen extensions, the group \(C_{\Bbbk ,2} \) and to study the capitulation problem of the 2-classes of \(\Bbbk \) .     

Hölder continuity for continuous solutions of the singular minimal surface equation with arbitrary zero set

In this paper we prove the following theorem: Let \(\Omega \subset \mathbb {R}^{n}\) be a bounded open set, \(\psi \in C_{c}^{2}(\mathbb {R}^{n})\), \(\psi > 0\) on \(\partial \Omega \), be given boundary values and u a nonnegative solution to the problem

$$\begin{aligned}&u \in C^{0}(\overline{\Omega }) \cap C^{2}(\{u> 0\}) \\&u = \psi \quad \text { on } \; \partial \Omega \\&{\text {div}} \left( \frac{Du}{\sqrt{1 + |Du|^{2}}}\right) = \frac{\alpha }{u \sqrt{1 + |Du|^{2}}} \quad \text { in } \; \{u > 0\} \end{aligned}$$

where \(\alpha > 0\) is a given constant. Then \(u \in C^{0, \frac{1}{2}} (\overline{\Omega })\). Furthermore we prove strict mean convexity of the free boundary \(\partial \{u = 0\}\) provided \(\partial \{u = 0\}\) is assumed to be of class \(C^{2}\) and \(\alpha \ge 1\).

     

Semipositive matrices and their semipositive cones

The semipositive cone of \(A\in \mathbb {R}^{m\times n}, K_A = \{x\ge 0\,:\, Ax\ge 0\}\), is considered mainly under the assumption that for some \(x\in K_A, Ax>0\), namely, that A is a semipositive matrix. The duality of \(K_A\) is studied and it is shown that \(K_A\) is a proper polyhedral cone. The relation among semipositivity cones of two matrices is examined via generalized inverse positivity. Perturbations and intervals of semipositive matrices are discussed. Connections with certain matrix classes pertinent to linear complementarity theory are also studied.

     

On Banach–Mazur distance between planar convex bodies

Upper estimates of the diameter and the radius of the family of planar convex bodies with respect to the Banach–Mazur distance are obtained. Namely, it is shown that the diameter does not exceed \(\tfrac{19-\sqrt{73}}{4}\approx 2.614\), which improves the previously known bound of 3, and that the radius does not exceed \(\frac{117}{70}\approx 1.671\).     

Ladders and canonical varieties of completely regular semigroups

A completely regular semigroup is a (disjoint) union of its (maximal) subgroups. We consider it here with the unary operation of inversion within its maximal subgroups. Their totality \(\mathcal {C}\mathcal {R}\) forms a variety whose lattice of subvarieties is denoted by \(\mathcal {L}(\mathcal {C}\mathcal {R})\). On it, one defines the relations \(\mathbf {B}^\wedge \) and \(\mathbf {B}^\vee \) by

$$\begin{aligned} \begin{array}{lll} \mathcal {U}\ \mathbf {B}^\wedge \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\cap \mathcal {B} =\mathcal {V}\cap \mathcal {B}, \\ \mathcal {U}\ \mathbf {B}^\vee \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\vee \mathcal {B} =\mathcal {V}\vee \mathcal {B} , \end{array} \end{aligned}$$

respectively, where \(\mathcal {B}\) denotes the variety of all bands. This is a study of the interplay between the \(\cap \)-subsemilatice \(\triangle \) of \(\mathcal {L}(\mathcal {C}\mathcal {R})\) of upper ends of \(\mathbf {B}^\wedge \)-classes and their \(\mathbf {B}^\vee \)-classes. The main tool is the concept of a ladder and their \(\mathbf {B}^\vee \)-classes, an indispensable part of the important Polák’s theorem providing a construction for the join of varieties of completely regular semigroups. The paper includes the tables of ladders of the upper ends of most \(\mathbf {B}^\wedge \)-classes. Canonical varieties consist of two ascending countably infinite chains which generate most of the upper ends of \(\mathbf {B}^\wedge \)-classes.

     

Idempotent and <Emphasis Type="Italic">p</Emphasis>-potent quadratic functions: distribution of nonlinearity and co-dimension

The Walsh transform \(\widehat{Q}\) of a quadratic function \(Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p\) satisfies \(|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}\) for all \(b\in {\mathbb F}_{p^n}\), where \(0\le s\le n-1\) is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class \(\mathcal {C}_1\) is defined for arbitrary n as \(\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}\), and the larger class \(\mathcal {C}_2\) is defined for even n as \(\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}\). For an odd prime p, the subclass \(\mathcal {D}\) of all p-ary quadratic functions is defined as \(\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}\). We determine the generating function for the distribution of the parameter s for \(\mathcal {C}_1, \mathcal {C}_2\) and \(\mathcal {D}\). As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case \(p > 2\), the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to \(\mathcal {C}_1\) and \(\mathcal {C}_2\) in terms of a generating function.     

Anderson Polymer in a Fractional Brownian Environment: Asymptotic Behavior of the Partition Function

We consider the Anderson polymer partition function

$$\begin{aligned} u(t):=\mathbb {E}^X\left[ e^{\int _0^t \mathrm {d}B^{X(s)}_s}\right] \,, \end{aligned}$$

where \(\{B^{x}_t\,;\, t\ge 0\}_{x\in \mathbb {Z}^d}\) is a family of independent fractional Brownian motions all with Hurst parameter \(H\in (0,1)\), and \(\{X(t)\}_{t\in \mathbb {R}^{\ge 0}}\) is a continuous-time simple symmetric random walk on \(\mathbb {Z}^d\) with jump rate \(\kappa \) and started from the origin. \(\mathbb {E}^X\) is the expectation with respect to this random walk. We prove that when \(H\le 1/2\), the function u(t) almost surely grows asymptotically like \(e^{\lambda t}\), where \(\lambda >0\) is a deterministic number. More precisely, we show that as t approaches \(+\infty \), the expression \(\{\frac{1}{t}\log u(t)\}_{t\in \mathbb {R}^{>0}}\) converges both almost surely and in the \(\hbox {L}^1\) sense to some positive deterministic number \(\lambda \). For \(H>1/2\), we first show that \(\lim _{t\rightarrow \infty } \frac{1}{t}\log u(t)\) exists both almost surely and in the \(\hbox {L}^1\) sense and equals a strictly positive deterministic number (possibly \(+\infty \)); hence, almost surely u(t) grows asymptotically at least like \(e^{\alpha t}\) for some deterministic constant \(\alpha >0\). On the other hand, we also show that almost surely and in the \(\hbox {L}^1\) sense, \(\limsup _{t\rightarrow \infty } \frac{1}{t\sqrt{\log t}}\log u(t)\) is a deterministic finite real number (possibly zero), hence proving that almost surely u(t) grows asymptotically at most like \(e^{\beta t\sqrt{\log t}}\) for some deterministic positive constant \(\beta \). Finally, for \(H>1/2\) when \(\mathbb {Z}^d\) is replaced by a circle endowed with a Hölder continuous covariance function, we show that \(\limsup _{t\rightarrow \infty } \frac{1}{t}\log u(t)\) is a deterministic finite positive real number, hence proving that almost surely u(t) grows asymptotically at most like \(e^{c t}\) for some deterministic positive constant c.

     

Explicit conformally constrained Willmore minimizers in arbitrary codimension

In our previous work (Ndiaye and Schätzle, 2014), we proved that the flat constant mean curvature tori $$\begin{aligned} T_r := r S^1 \times \sqrt{1 - r^2} S^1 \subseteq S^3 \quad \hbox {for } 0 < r \le 1/\sqrt{2} \end{aligned}$$ minimize the Willmore energy in their conformal class in codimension one when \(r \approx 1 / \sqrt{2}\) , that is \(T_r\) is close to the Clifford torus \(T_{Cliff} = T_{1/\sqrt{2}}\) . In this article, we extend this to arbitrary codimension. Moreover we prove that the Clifford torus minimizes the Willmore energy in an open neighbourhood of its conformal class, again in arbitrary codimension, but the neighbourhood may depend on the codimension.