A translating soliton in Euclidean space R3 is a surface whose mean curvature H satisfies H=−⟨N,v→⟩, where N is the unit normal of the surface and v→∈R3 is a fixed unit vector. In this note, we prove that any translating soliton that can be expressed as the set of zeros of an implicit equation of type f(x)+g(y)+h(z)=0 in Cartesian coordinates, where f, g, and h are smooth real functions of one variable, must be a tilted grim reaper or a rotational surface.
Given a symplectic (respectively, orthogonal) parabolic vector bundle over a compact Riemann surface, we prove that its pullback and direct image through a map between compact Riemann surfaces inherit a natural symplectic (respectively, orthogonal) structure. If the parabolic bundle is endowed with a parabolic Higgs field or a parabolic connection which are compatible with the symplectic (respectively, orthogonal) structure, then its pullback and direct image are also compatible with the resulting symplectic (respectively, orthogonal) structure. We also show that these constructions are preserved through the Nonabelian Hodge Correspondence.
We explore the dependence of the minimal integral Mahler measure of Galois quartic fields on the discriminant of the field. We obtain density results which are conditional on the ABC conjecture as well as several unconditional results.
We introduce a generalization of the Bourgain-Rosenthal-Schechtman R_ω^p space: Let Y be a Haar system Hardy space, i.e., a separable rearrangement-invariant function space on the unit interval or an associated Hardy space defined via the square function (such as dyadic H^1). Then we define Y_ω as the closed linear span in Y of independent distributional copies of the spaces Y_n of dyadic step functions at scale 2^-n. Combining finite-dimensional and infinite-dimensional techniques, we prove that the identity operator I on Y_ω factors through every bounded linear operator T on Y_ω which has large diagonal, and in general, the identity factors either through T or through I - T.
The terms "whiskering", and more generally "grafting", refer to adding generators to any monomial ideal to make the resulting ideal Cohen-Macaulay. We investigate the independence complexes of simplicial complexes that are constructed through a whiskering or grafting process, and we show that these independence complexes are (generalized) Bier balls. More specifically, the independence complexes are either homeomorphic to a ball or a sphere. In a related direction, we classify when the independence complexes of very well-covered graphs are homeomorphic to balls or spheres.
We consider Euclidean lattices spanned by images of algebraic conjugates of an algebraic number under Minkowski embedding, investigating their rank, properties of their automorphism groups and sets of minimal vectors. We are especially interested in situations when the resulting lattice is well-rounded. We show that this happens for large Pisot numbers of prime degree, demonstrating infinite families of such lattices. We also fully classify well-rounded lattices from algebraic conjugates in the 2-dimensional case and present various examples in the 3-dimensional case. Finally, we derive a determinant formula for the resulting lattice in the case when the minimal polynomial of an algebraic number has its Galois group of a particular type.
In this article, we define and explore the weak normalization of an affine semigroup. In particular, for a fixed prime integer, we provide a geometric description of the weak normalization of an affine semigroup with respect to that prime, which corresponds to the weak normalization of the affine semigroup ring over a field of that prime characteristic, similar to the description of the seminormalization of an affine semigroup given by Reid-Roberts. We then use this description to understand the singularities of an affine semigroup ring defined over a field of prime characteristic and provide several examples. In particular, we demonstrate that all affine semigroup rings defined over fields of prime characteristic have a uniform upper bound on the Frobenius test exponent of all ideals, which provides a large and important class of examples with a positive answer to a question of Katzman-Sharp on uniformity of Frobenius test exponents. Finally, we provide an algorithm and implementation to compute the weak normalization of an affine semigroup, as well as the Frobenius test exponent and Frobenius closures of ideals in the affine semigroup ring.
Compact Kähler manifolds classically satisfy the Hard Lefschetz Theorem, which gives strong control on the underlying topology of the manifold. One expects a similar theorem to be true for Kähler Lie Algebroids, and we show for a certain class of them that this is indeed true, with an added ellipticity requirement. We provide examples of Lie Algebroids satisfying this, as well as an example of a Kähler Lie Algebroid that does not meet this Ellipticity requirement, and consequently fails to satisfy the Hard Lefschetz condition.
Shifts of finite type defined from shift equivalent matrices must be flow equivalent.
In this article, we study the essential pseudospectra via polynomially strict singular operators, which generalizes the class of strict singular operators. We present some new results in essential pseudospectra for closed linear operators in a Banach space under perturbations by polynomially strict singular operators. Furthermore, we apply the obtained results to analyze the incidence of some perturbation results on left (resp. right) Weyl essential pseudospectra and left (resp. right) Fredholm essential pseudospectra. In addition, we will describe the essential pseudospectra of a sum of two bounded linear operators. A final application of the obtained results is to characterize the pseudo-left (resp. pseudo-right) Fredholm spectra of 2 x 2 block operator matrices.
On a Hilbert space H, a dissipative C0 contraction semigroup T and the discrete semigroup, D, of its cogenerator T split the space into recurrent and weakly stable subspaces. The pair (T , T) defines three limit algebras K that are maximal abelian, self adjoint, and W* and define the dynamics of T ; D. Each K has a recurrent space that is the span of recurrent limit cycles defined by K on the initial states. The pair T ; D has a common weakly stable space Hwiff they have common recurrent space Hm and limit algebra M Omega that is maximal abelian self adjoint (m.a.s.a.) and defines the limit cycles for both.
Given the norms of powers .kxnk/n >= o of a Banach algebra element x, the largest possible value of the minimum modulus on the spectrum of x is determined. It is also shown that, given a Banach algebra element x and a compact set K subset of & Copf; with maximum modulus no more than the spectral radius of x, there exists a Banach algebra element y with kynk D kxnk for all n >= 0 and spectrum equal to the union of the spectrum of x and K. These results, along with the spectral radius formula, are generalized to the joint spectrum of several commutative Banach algebra elements. The generalization of the spectral radius formula presented gives the maximum possible joint spectrum for commutative Banach algebra elements xi; ::: ; xn, given the norms .kxi
Our main goal is to prove that the space of weighted vanishing mean oscillation VMO(w) is the predual of the weighted Hardy space H1w(Rn), 1 w ( R n ), for w 2 A1. 1. In order to do this we, will use the theory of weighted tent spaces and its connection with the Hardy space and VMO(w).
We explore the Bohr inequality involving the Fourier transforms of complex-valued integrable and square integrable functions defined on a second countable compact topological group. We also investigate the connection of the Bohr phenomenon with a modulus of convexity of the space of bounded linear operators defined on a complex Hilbert space.
This paper investigates whether two independent elephant random walks (ERWs) on & Zopf;, each with a different memory parameter, can meet infinitely often, extending the work of Roy, Takei, and Tanemura. We also study the asymptotic behavior of their distance by providing an elementary and accessible proof of the classical law of the iterated logarithm (LIL) for centered, continuous, self-similar Gaussian processes under a certain decay condition on the covariance kernel.
We use the lens of Zappa-Sz & eacute;p decomposition to examine the relationship between directed graph products and k-graph products. There are many examples of higher-rank graphs, or k-graphs, whose underlying directed graph may be factored as a product, but the k-graph itself is not a product. In such examples, we establish that the Zappa-Sz & eacute;p structure of the k-graph gives rise to "actions" of the underlying directed factors on each other. Although these "actions" are in general poorly behaved, if one of them is trivial (or trivial up to isomorphism), we obtain a crossed-product-like structure on the k-graph. We provide examples where this crossed-product structure is visible in the associated C*-algebra, and we characterize those k-graphs whose Zappa-Sz & eacute;p induced actions are trivial up to isomorphism.
In this article, we study the similarity of the Polish operator topologies WOT, SOT, SOT & lowast;, and SOT & lowast; on the set of the positive contractions on & ell;p with p>1. Using the notion of norming vector for a positive operator, we prove that these topologies are similar on P1(& ell;2); that is, they have the same dense sets in P1(& ell;2). In particular, these topologies will share the same comeager sets in P1(& ell;2). We then apply these results to the study of typical properties of positive contractions on & ell;p-spaces in the Baire category sense. In particular, we prove that a typical positive contraction T is an element of(P1(& ell;2),SOT) has no eigenvalue. This stands in strong contrast to a result of Eisner and M & aacute;trai, stating that the point spectrum of a typical contraction T is an element of(B1(& ell;2),SOT) contains the whole unit disk. As a consequence of our results, we obtain that a typical positive contraction T is an element of(P1(& ell;2),WOT) (resp. T is an element of(P1(& ell;2),SOT & lowast;)) has no eigenvalue
