In this paper, we study a special class of interval translation mappings introduced in 2007 by H. Bruin. We prove that a typical map of infinite type is uniquely ergodic and provide an upper bound for the Hausdorff dimension of the parameter set that gives rise to the interval translation mappings of infinite type. Moreover, we prove that such interval translation mappings have sublinear combinatorial complexity.
The existence of weak solutions to the initial–boundary value problem for the mathematical model describing the motion of a nonlinearly elastically retarded Navier–Stokes–Voigt fluid is studied in this paper. In this model, the fluid viscosity is considered as a nonlinear function. In addition, the temperature is also taken into account, which leads to the emergence of an additional energy balance equation. The proof is based on the topological approximation approach to the study of hydrodynamic problems, as well as the following iterative process.
Some properties of the weak homological bidimension of Banach algebras are studied and important examples of its calculation are given. In particular, this characteristic is calculated for all semisimple biprojective Banach algebras with the approximation property, all so-called tensor algebras generated by bilinear forms, and all infinite-dimensional Hilbert algebras. In addition, the additivity formula for weak bidimension is proved and it is shown that, in the class of semisimple Banach algebras, this homological characteristic can take any natural values, as well as the values 0 and ∞ .
We consider complete seminormed spaces of functions of one real variable whose seminorm has a finite-dimensional kernel. If the seminorm is invariant under affine changes of the argument, we call such a space interesting. We prove that the maximal interesting space embedded in L_1,loc(ℝ^n) is equivalent to BMO , and the maximal interesting space embedded in 𝒟'(ℝ) is equivalent to the homogeneous Besov space Ḃ^0_∞,∞ . We also construct a minimal interesting space that contains the space of smooth functions with compact support.
We develop a shearlet expansion theory for the Lizorkin-type spaces 𝒮_0(ℝ^2) and 𝒮'_0(ℝ^2) . We prove that the shearlet series expansion with respect to a Parseval shearlet converges in the topology of these spaces and provide a topological characterization of the Lizorkin space of distributions in terms of shearlet coefficients. Finally, we apply our distributional shearlet expansion theory to analyze asymptotic properties of distributions and obtain several Tauberian-type results that characterize the quasiasymptotics and quasiasymptotically boundedness of Lizorkin distributions via the asymptotic behavior of their shearlet coefficients.
The construction of ordinary commuting differential operators is a classical problem of differential equations and integrable systems, which has applications in soliton theory. Commuting operators of rank 1 were found by Krichever. The problem of constructing operators of rank l>1 has not been solved in the general case. In all known examples of operators of rank l>1 , the spectral curves are hyperelliptic curves. In this paper, the first examples of operators of rank 2, corresponding to trigonal spectral curves of genus 3, are constructed.
Let 𝒜 be a complex unital commutative Banach algebra. Let φ𝒜→ℂ be a map such that for x,y∈𝒜 , φ(x)-φ(y)∈σ_ε(x-y) and φ has ℂ -linear differentials almost everywhere. Then φ is approximately multiplicative. A similar conclusion is reached by replacing the differential condition with comparable assumptions on the map. This result is similar to the Kowalski–Słodkowski theorem. Analogous versions of it are also discussed for the exponential spectrum and for a particular class of the Ransford spectrum.
To every finite word of a finite alphabet 𝐀⊆ℕ , one can add a prefix V and a suffix W that are fixed finite words of the alphabet ℕ . The words thus obtained are related to finite continued fraction expansions of certain numbers from (0,1)∩ℚ . Their irreducible denominators that belong to [1,N] form the set 𝔇^N_𝐀,V,W . In the author’s previous work, it was shown that, under certain conditions on 𝐀 and Q∈ℕ , and provided that the lengths of V and W are not very large, the set 𝔇^N_𝐀,V,W contains almost all possible remainders modulo Q ; moreover, the corresponding formula has power-law decay. In this paper, we obtain an analogous formula for an arbitrarily long suffix W .
In L_2(ℝ^d) , we consider a self-adjoint operator which is the sum of a convolution operator and a potential. With minimal assumptions on the convolution kernel and the potential, we describe the location of its essential spectrum and give sufficient conditions for the existence of infinite series of discrete eigenvalues accumulating at the edges of the essential spectrum. We also discuss the case where a non-empty discrete spectrum appears in gaps of the essential spectrum.
This paper first demonstrates the existence and uniqueness of solutions to homogeneous Dirichlet boundary value problems for second-order linear elliptic equations with L^2 -drifts of negative divergence and positive L^1 -zero-order terms, based on a functional analytic approach, including weak convergence methods and duality arguments. By improving the previous contraction properties, which may not be effective when the zero-order term is very small, this paper introduces a general L^2 -“contraction” property for any positive zero-order term, leading to remarkable results regarding L^2 -stability. These stability results are applicable to L^2 -error analysis for physics-informed neural networks, and can also be applied to stationary Schrödinger operators with L^2 -zero-order terms. We emphasize that all the constants arising in the estimates of this paper can be explicitly computed.
We study a two-dimensional massive Dirac operator with a singular potential supported on a periodic graph, and examine the self-adjointness and the Fredholmness of the associated unbounded operator.
Bohl points of a conditionally periodic motion are defined as the phases such that the integral of a continuous function with zero mean value along the motion is always nonnegative (or nonpositive). Bohl points are known to always exist. This note is devoted to a generalization of this result to the case of uniquely ergodic dynamical systems as well as to almost periodic Bohr functions.
We construct an example of a field and a smooth del Pezzo surface of degree 2 over this field without points such that its automorphism group is isomorphic to PSL_2(𝔽_7) ×ℤ/2ℤ , which is the largest possible automorphism group for del Pezzo surfaces of degree 2 over an algebraically closed field of characteristic zero.
Veselý (1997) studied Banach spaces that admit f -centers for finite subsets of the space. In this work, we introduce the concept of the ℱ -simultaneous approximative τ -compactness property ( τ - ℱ - SACP or SACP for short) for triplets (X, V,𝔉) , where X is a Banach space, V is a τ -closed subset of X , 𝔉 is a subfamily of closed and bounded subsets of X , ℱ is a collection of functions, and τ is the norm or weak topology on X . We characterize reflexive spaces with the Kadec–Klee property using triplets with τ - ℱ - SACP . We investigate the relationship between τ - ℱ - SACP and the continuity properties of the restricted f -center map. The study further examines τ - ℱ - SACP in the context of CLUR -spaces and explores various characterizations of τ - ℱ - SACP , including connections to reflexivity, Fréchet smoothness, and the Kadec–Klee property.
We investigate birational properties of hypersurfaces of degree 6 in the weighted projective space ℙ(1,1,2,2,3) . In particular, we prove that any such quasi-smooth hypersurface is not rational.
A rational function on a real algebraic curve C is called separating if it takes real values only at real points. Such a function defines a covering ℝ C→ℝℙ^1 . Let c_1,…,c_r be the connected components of ℝ C . M. Kummer and K. Shaw defined the separating semigroup of C as the set of all sequences (d_1(f),…,d_r(f)) where f is a separating function, and d_i(f) is the degree of the restriction of f to c_i . In the present paper, we describe the separating semigroups of all genus 4 curves. For the proofs, we consider the canonical embedding of C into a quadric X in ℙ^3 , and apply Abel’s theorem to 1-forms on C obtained as Poincaré residues of certain meromorphic 2-forms.
We prove that the number of dissections of a given polygon into triangles with fixed areas of faces is finite and that an equidissection is algebraic as long as the vertices of the original polygon have algebraic coordinates.
A triangulation of a circle bundle $ E \xrightarrow[\text{}]{\pi} B$ is a triangulation of $E$ and $B$ such that $\pi$ is a simplicial map. In the paper we address the following questions: Which circle bundles can be triangulated over a given triangulation of the base? What are the minimal triangulations of a bundle? A complete solution for semisimplicial triangulations was given by N. Mn\"{e}v. Our results deals with classical triangulations, that is, simplicial complexes. We give an exact answer for a wide family of triangulated spheres (including the boundary of the $3$-simplex, the boundary of the octahedron, the suspension over an $n$-gon, icosahedron). In the general case we present a sufficient criterion for existence of a triangulation. Some minimality results follow straightforwadly.
We investigate birational properties of hypersurfaces of degree $$6$$ in the weighted projective space $$\mathbb{P}(1,1,2,2,3)$$ . In particular, we prove that any such quasi-smooth hypersurface is not rational.
