Matrix near-rings and 0-primitivity

In this article, we study the implication of the primitivity of a matrix near-ring ${\mathbb{M}_n(R) (n >1 )}$ and that of the underlying base near-ring R. We show that when R is a zero-symmetric near-ring with identity and ${\mathbb{M}_n(R)}$ has the descending chain condition on ${\mathbb{M}_n(R)}$ -subgroups, then the 0-primitivity of ${\mathbb{M}_n(R)}$ implies the 0-primitivity of R. It is not known if this is true when the descending chain condition on ${\mathbb{M}_n(R)}$ is removed. On the other hand, an example is given to show that this is not true in the case of generalized matrix near-rings.     

The affine preservers of non-singular matrices

When \mathbbK{\mathbb{K}} is an arbitrary field, we study the affine automorphisms of M_n(\mathbbK){{\rm M}_n(\mathbb{K})} that stabilize GL_n(\mathbbK){{\rm GL}_n(\mathbb{K})}. Using a theorem of Dieudonné on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n > 2 or # ${\mathbb{K} > 2}${\mathbb{K} > 2}. We include a short new proof of the more general Flanders theorem for affine subspaces of M_p,q(\mathbbK){{\rm M}_{p,q}(\mathbb{K})} with bounded rank. We also find that the group of affine transformations of M₂(\mathbbF₂){{\rm M}_2(\mathbb{F}_2)} that stabilize GL₂(\mathbbF₂){{\rm GL}_2(\mathbb{F}_2)} does not consist solely of linear maps. Using the theory of quadratic forms over \mathbbF₂{\mathbb{F}_2}, we construct explicit isomorphisms between it, the symplectic group Sp₄(\mathbbF₂){{\rm Sp}_4(\mathbb{F}_2)} and the symmetric group \mathfrakS₆{\mathfrak{S}_6}.     

Metrizability of the space of {\mathbb{R}} -places of a real function field

For n = 1, the space of ${\mathbb{R}}For n = 1, the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x₁,?, x_n){\mathbb{R}(x_1,\ldots, x_n)} is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x, y){\mathbb{R}(x, y)} is not metrizable. We explain how the proof generalizes to show that the space of \mathbbR{\mathbb{R}} -places of any finitely generated formally real field extension of \mathbbR{\mathbb{R}} of transcendence degree ≥ 2 is not metrizable. We also consider the more general question of when the space of \mathbbR{\mathbb{R}} -places of a finitely generated formally real field extension of a real closed field is metrizable.     

Homomorphisms with respect to a function

Let X be a realcompact space and H:C(X)?\mathbbR{H:C(X)\rightarrow\mathbb{R}} be an identity and order preserving group homomorphism. It is shown that H is an evaluation at some point of X if and only if there is j ? C(\mathbbR){\varphi\in C(\mathbb{R})} with ${\varphi(r)>\varphi(0)}${\varphi(r)>\varphi(0)} for all r ? \mathbbR-{0}{r\in\mathbb{R}-\{0\}} for which H°j = j°H{H\circ\varphi=\varphi\circ H} . This extends (and unifies) classical results by Hewitt and Shirota.     

Spectral theorem for multipliers on {L_{\omega}^{2}(\mathbb{R})}

We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces L_w²(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with f ? C_c(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces L²_w(\mathbb R^k){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form.     

On some inequalities for Doob decompositions in Banach function spaces

Let ${\Phi : \mathbb{R} \to [0, \infty)}Let F: \mathbbR ? [0, ￥){\Phi : \mathbb{R} \to [0, \infty)} be a Young function and let f = (f_n)_n ? \mathbbZ₊{f = (f_n)_n\in\mathbb{Z}_{+}} be a martingale such that F(f_n) ? L₁{\Phi(f_n) \in L_1} for all n ? \mathbbZ₊{n \in \mathbb{Z}_{+}} . Then the process F(f) = (F(f_n))_n ? \mathbbZ₊{\Phi(f) = (\Phi(f_n))_n\in\mathbb{Z}_{+}} can be uniquely decomposed as F(f_n)=g_n+h_n{\Phi(f_n)=g_n+h_n} , where g=(g_n)_n ? \mathbbZ₊{g=(g_n)_n\in\mathbb{Z}_{+}} is a martingale and h=(h_n)_n ? \mathbbZ₊{h=(h_n)_n\in\mathbb{Z}_{+}} is a predictable nondecreasing process such that h ₀ = 0 almost surely. The main results characterize those Banach function spaces X such that the inequality ||h_￥||_X ￡ C ||F(Mf) ||_X{\|{h_{\infty}}\|_{X} \leq C \|{\Phi(Mf)} \|_X} is valid, and those X such that the inequality ||h_￥||_X ￡ C ||F(Sf) ||_X{\|{h_{\infty}}\|_{X} \leq C \|{\Phi(Sf)} \|_X} is valid, where Mf and Sf denote the maximal function and the square function of f, respectively.     

A Nonlocal Diffusion Equation whose Solutions Develop a Free Boundary

Let J:\mathbbR ? \mathbbRJ:\mathbb{R} \to \mathbb{R} be a nonnegative, smooth compactly supported function such that ò_\mathbbR J(r)dr = 1. \int_\mathbb{R} {J(r)dr = 1.} We consider the nonlocal diffusion problem

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Let C( \mathbbR^m ) C\left( {{\mathbb{R}^m}} \right) be the space of bounded and continuous functions x:\mathbbR^m ? \mathbbR x:{\mathbb{R}^m} \to \mathbb{R} equipped with the norm

Weakly Singular and Microscopically Hypersingular Load Perturbation for a Nonlinear Traction Boundary Value Problem: a Functional Analytic Approach

Let Ω ⁱ and Ω ^o be two bounded open subsets of \mathbbRⁿ{{\mathbb{R}}^{n}} containing 0. Let G ⁱ be a (nonlinear) map from ?Wⁱ×\mathbbRⁿ{\partial\Omega^{i}\times {\mathbb{R}}^{n}} to \mathbbRⁿ{{\mathbb{R}}^{n}} . Let a ^o be a map from ∂Ω ^o to the set M_n(\mathbbR){M_{n}({\mathbb{R}})} of n × n matrices with real entries. Let g be a function from ∂Ω ^o to \mathbbRⁿ{{\mathbb{R}}^{n}} . Let γ be a positive valued function defined on a right neighborhood of 0 in the real line. Let T be a map from ]1-(2/n),+￥[×M_n(\mathbbR){]1-(2/n),+\infty[\times M_{n}({\mathbb{R}})} to M_n(\mathbbR){M_{n}({\mathbb{R}})} . Then we consider the problem

Balanced Deep Matrix Algebras

This paper continues the study of associative and Lie deep matrix algebras, DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) and \mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras, BDM(X,\mathbbK){\mathcal{BDM}}(X,{\mathbb{K}}) and \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}), are defined and studied for an infinite set X. The global structures of these two algebras are studied, devising a depth grading on both as well as determining their ideal lattices. In particular, \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) is shown to be semisimple. The Lie algebra \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) possesses a deep Cartan decomposition and is locally finite with every finite subalgebra naturally enveloped by a semi-direct product of \mathfraksl_n{\mathfrak{{sl}_n}}’s. We classify all associative bilinear forms on \mathfraksl₂\mathfrakd{\mathfrak{sl}_2\mathfrak{d}} (a natural depth analogue of \mathfraksl₂{\mathfrak{{sl}_2}}) and \mathfrakbld{\mathfrak{bld}}.     

Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$

We prove the existence of a global heat flow u : Ω ×  \mathbbR⁺ ? \mathbbR^N {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR⁺ {\mathbb{R}^{+}}) ⊂  \mathbbRⁿ {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbR^N {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbR^N {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.     

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

Teichmüller Spaces and Negatively Curved Fiber Bundles

In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map F _X between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres ${\mathbb{S}^k}In the first part of the paper we introduce the theory of bundles with negatively curved fibers. For a space X there is a forgetful map F _X between bundle theories over X, which assigns to a bundle with negatively curved fibers over X its subjacent smooth bundle. Our main result states that, for certain k-spheres \mathbbS^k{\mathbb{S}^k}, the forgetful map F_\mathbbS^k{F_{\mathbb{S}^k}} is not one-to-one. This result follows from Theorem A, which proves that the quotient map MET  ^{sec < 0} (M)?T  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} is not trivial at some homotopy levels, provided the hyperbolic manifold M satisfies certain conditions. Here MET  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)} is the space of negatively curved metrics on M and T  ^{sec < 0} (M) = MET  ^{sec < 0} (M)/ DIFF₀(M){\mathcal{T}^{\,\,sec <0 }(M) = \mathcal{MET}^{\,\,sec <0 }(M)/ {\rm DIFF}_0(M)} is, as defined in [FO2], the Teichmüller space of negatively curved metrics on M. In particular we conclude that T  ^{sec < 0} (M){\mathcal{T}^{\,\,sec <0 }(M)} is, in general, not connected. Two remarks: (1) the nontrivial elements in p_kMET  ^{sec < 0} (M){\pi_{k}\mathcal{MET}^{\,\,sec <0 }(M)} constructed in [FO3] have trivial image by the map induced by MET  ^{sec < 0} (M)?T  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} ; (2) the nonzero classes in p_kT  ^{sec < 0} (M){\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)} constructed in [FO2] are not in the image of the map induced by MET  ^{sec < 0} (M)?T  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)\rightarrow\mathcal{T}^{\,\,sec <0 }(M)} ; the nontrivial classes in p_kT  ^{sec < 0} (M){\pi_{k}\mathcal{T}^{\,\,sec <0 }(M)} given here, besides coming from MET  ^{sec < 0} (M){\mathcal{MET}^{\,\,sec <0 }(M)} and being harder to construct, have a different nature and genesis: the former classes – given in [FO2] – come from the existence of exotic spheres, while the latter classes – given here – arise from the non-triviality and structure of certain homotopy groups of the space of pseudo-isotopies of the circle \mathbbS¹{\mathbb{S}^1}. The strength of the new techniques used here allowed us to prove also a homology version of Theorem A, which is given in Theorem B.     

Optimal Embeddings of Spaces of Generalized Smoothness in the Critical Case

We study necessary and sufficient conditions for embeddings of Besov and Triebel-Lizorkin spaces of generalized smoothness B^(n/p,Y)_p,q(\mathbbRⁿ)B^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n}) and F^(n/p,Y)_p,q(\mathbbRⁿ)F^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n}), respectively, into generalized H?lder spaces L_￥,r^m(·)( \mathbb Rⁿ)\Lambda_{\infty,r}^{\mu(\cdot)}(\ensuremath {\ensuremath {\mathbb {R}}^{n}}). In particular, we are able to characterize optimal embeddings for this class of spaces provided q>1. These results improve the embedding assertions given by the continuity envelopes of B^(n/p,Y)_p,q(\mathbbRⁿ)B^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n}) and F^(n/p,Y)_p,q(\mathbbRⁿ)F^{(n/p,\Psi)}_{p,q}(\mathbb{R}^{n}), which were obtained recently solving an open problem of D.D. Haroske in the classical setting.     

On the rationality of the moduli space of Lüroth quartics

We prove that the moduli space \mathfrakM_L{\mathfrak{M}_L} of Lüroth quartics in \mathbbP²{\mathbb{P}^2}, i.e. the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of PGL₃ (\mathbbC){\mathrm{PGL}_3 (\mathbb{C})} is rational, as is the related moduli space of Bateman seven-tuples of points in \mathbbP²{\mathbb{P}^2}.     

Orthogonal basis for spherical monogenics by step two branching

Spherical monogenics can be regarded as a basic tool for the study of harmonic analysis of the Dirac operator in Euclidean space \mathbb R^m{{\mathbb R}^m}. They play a similar role as spherical harmonics do in case of harmonic analysis of the Laplace operator on \mathbb R^m{{\mathbb R}^m}. Fix the direct sum \mathbb R^m=\mathbb R^p ?\mathbb R^q{{\mathbb R}^m={\mathbb R}^p \oplus {\mathbb R}^q}. In this article, we will study the decomposition of the space M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)} of spherical monogenics of order n under the action of Spin(p) × Spin(q). As a result, we obtain a Spin(p) × Spin(q)-invariant orthonormal basis for M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}. In particular, using the construction with p = 2 inductively, this yields a new orthonormal basis for the space M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}.     

Solutions to Polynomial Generalized Bers–Vekua Equations in Clifford Analysis

In this paper, we mainly study polynomial generalized Vekua-type equation _boxclose)w=0{p(\mathcal{D})w=0} and polynomial generalized Bers–Vekua equation p(D)w=0{p(\mathcal{\underline{D}})w=0} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}} where D{\mathcal{D}} and D{\mathcal{\underline{D}}} mean generalized Vekua-type operator and generalized Bers–Vekua operator, respectively. Using Clifford algebra, we obtain the Fischer-type decomposition theorems for the solutions to these equations including (D-l)^kw=0,(D-l)^kw=0(k ? \mathbbN){\left(\mathcal{D}-\lambda\right)^{k}w=0,\left(\mathcal {\underline{D}}-\lambda\right)^{k}w=0\left(k\in\mathbb{N}\right)} with complex parameter λ as special cases, which derive the Almansi-type decomposition theorems for iterated generalized Bers–Vekua equation and polynomial generalized Cauchy–Riemann equation defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}. Making use of the decomposition theorems we give the solutions to polynomial generalized Bers–Vekua equation defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}} under some conditions. Furthermore we discuss inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}, and develop the structure of the solutions to inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}.