Dimensional Theorem Research Articles

A proof to direct sum decomposition for finite dimension of linear space

In higher algebra, Direct sum decomposition theorem for finite dimension of linear space is a classical result of linear transformation. This theorem has been widely used in many fields, such as algebra and mechanics and so on. In this paper, the proof is given by means of Hamilton Cayley theorem.

The Difference in Molecular Orientation and Interphase Structure of SiO2/Shape Memory Polyurethane in Original, Programmed and Recovered States during Shape Memory Process.

In order to further understand the shape memory mechanism of a silicon dioxide/shape memory polyurethane (SiO2/SMPU) composite, the thermodynamic properties and shape memory behaviors of prepared SiO2/SMPU were characterized. Dynamic changes in the molecular orientation and interphase structures of SiO2/SMPU during a shape memory cycle were then discussed according to the small angle X-ray scattering theory, Guinier’s law, Porod approximation, and fractal dimension theorem. In this paper, a dynamic mechanical analyzer (DMA) helped to determine the glass transition start temperature (Tg) by taking the onset point of the sigmoidal change in the storage modulus, while transition temperature (Ttrans) was defined by the peak of tan δ, then the test and the calculated results indicated that the Tg of SiO2/SMPU was 50.4 °C, and the Ttrans of SiO2/SMPU was 72.18 °C. SiO2/SMPU showed good shape memory performance. The programmed SiO2/SMPU showed quite obvious microphase separation and molecular orientation. Large-size sheets and long-period structures were formed in the programmed SiO2/SMPU, which increases the electron density difference. Furthermore, some hard segments had been rearranged, and their gyration radii decreased. In addition, several defects formed at the interfaces of SiO2/SMPU, which caused the generation of space charges, thus leading to local electron density fluctuations. The blurred interphase structure and the intermediate layer formed in the programmed SiO2/SMPU and there was evident crystal damage and chemical bond breakage in the recovered SiO2/SMPU. Finally, the original and recovered SiO2/SMPU samples belong to the surface fractal system, but the programmed sample belongs to the mass fractal and reforms two-phase structures. This study provides an insight into the shape memory mechanism of the SiO2/SMPU composite.

Hyper vector spaces over Krasner hyperfields

In this paper, we study hyper vector spaces over Krasner hyperfields. First, we introduce the notions of linearly independence (dependence) and basis for a hyper vector space. Second, we investigate their properties and prove some results for hyper vector spaces that are similar to that of vector spaces over fields. Then, we define linear transformations over hyper vector spaces and investigate their properties. Finally, we prove the dimension theorem for linear transformations.

Teichmüller’s Theorem in Higher Dimensions and Its Applications

For a given ring (domain) in \(\overline{\mathbb {R}}^n\), we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all \(n\ge 3\), the standard definition of uniformly perfect sets in terms of the Euclidean metric is equivalent to the boundedness of the moduli of the separating rings. We also establish separation theorems for a “half” of a ring. As applications of those results, we will prove boundary Hölder continuity of quasiconformal mappings of the ball or the half space in \(\mathbb {R}^n\).

Minimal semi-flat-cotorsion replacements and cosupport

Over a commutative noetherian ring R of finite Krull dimension, we show that every complex of flat cotorsion R-modules decomposes as a direct sum of a minimal complex and a contractible complex. Moreover, we define the notion of a semi-flat-cotorsion complex as a special type of semi-flat complex, and provide functorial ways to construct a quasi-isomorphism from a semi-flat complex to a semi-flat-cotorsion complex. Consequently, every R-complex can be replaced by a minimal semi-flat-cotorsion complex in the derived category over R. Furthermore, we describe structure of semi-flat-cotorsion replacements, by which we recover classic theorems for finitistic dimensions. In addition, we improve some results on cosupport and give a cautionary example. We also explain that semi-flat-cotorsion replacements always exist and can be used to describe the derived category over any associative ring.

Generalized Boundary Condition Applied to Lieb-Schultz-Mattis-Type Ingappabilities and Many-Body Chern Numbers

We introduce a new boundary condition which renders the flux-insertion argument for the Lieb-Schultz-Mattis type theorems in two or higher dimensions free from the specific choice of system sizes. It also enables a formulation of the Lieb-Schultz-Mattis type theorems in arbitrary dimensions in terms of the anomaly in field theories of $1+1$ dimensions with a bulk correspondence as a BF-theory in 2+1 dimensions. Furthermore, we apply the anomaly-based formulation to the constraints on a half-filled spinless fermion on a square lattice with $\pi$ flux, utilizing time-reversal, the magnetic translation and on-site internal $U(N)$ symmetries. This demonstrates the role of time-reversal anomaly on the ingappabilities of a lattice model.

Knapp–Stein dimension theorem for finite central covering groups

Knapp–Stein dimension theorem for finite central covering groups

On the Assouad dimension of projections

Let $F \subset \mathbb{R}^{2}$, and let $\dim_{\mathrm{A}}$ stand for Assouad dimension. I prove that $\dim_{\mathrm{A}} \pi_{e}(F) \geq \min\{\dim_{\mathrm{A}} F,1\}$ for all $e \in S^{1}$ outside of a set of Hausdorff dimension zero. This is a strong variant of Marstrand's projection theorem for Assouad dimension, whose analogue is not true for other common notions of fractal dimension, such as Hausdorff or packing dimension.

Soft theorems from compactification

We analyze the single subleading soft graviton theorem in (d + 1) dimensions under compactification on S1. This produces the single soft theorems for the graviton, vector and scalar fields in d dimension. For the compactification of 11-dimensional supergravity theory, this gives the soft factorization properties of the single graviton, dilaton and RR 1-form fields in type IIA string theory in ten dimensions. For the case of the soft vector field, we also explicitly check the result obtained from compactification by computing the amplitudes with external massive spin two and massless finite energy states interacting with soft vector field. The former are the Kaluza-Klein excitations of the d + 1 dimensional metric. Describing the interaction of the KK-modes with the vector field at each level by the minimally coupled Fierz-Pauli Lagrangian, we find agreement with the results obtained from the compactification if the gyromagnetic ratio in the minimally coupled Fierz-Pauli Lagrangian is taken to be g = 1.

A note on the stratification by automorphisms of smooth plane curves of genus 6

In this note, we give a so-called representative classification for the strata by automorphism group of smooth $\bar{k}$-plane curves of genus $6$, where $\bar{k}$ is a fixed separable closure of a field $k$ of characteristic $p = 0$ or $p > 13$. We start with a classification already obtained by the first author and we use standard techniques. Interestingly, in the way to get these families for the different strata, we find two remarkable phenomenons that did not appear before. One is the existence of a non $0$-dimensional final stratum of plane curves. At a first sight it may sound odd, but we will see that this is a normal situation for higher degrees and we will give a explanation for it. We explicitly describe representative families for all strata, except for the stratum with automorphism group $\mathbb{Z}/5\mathbb{Z}$. Here we find the second difference with the lower genus cases where the previous techniques do not fully work. Fortunately, we are still able to prove the existence of such family by applying a version of Luroth's theorem in dimension $2$.

Product-Mix Auctions and Tropical Geometry

In a recent and ongoing work, Baldwin and Klemperer explore a connection between tropical geometry and economics. They give a sufficient condition for the existence of competitive equilibrium in product-mix auctions of indivisible goods. This result, which we call the unimodularity theorem, can also be traced back to the work of Danilov, Koshevoy, and Murota in discrete convex analysis. We give a new proof of the unimodularity theorem via the classical unimodularity theorem in integer programming. We give a unified treatment of these results via tropical geometry and formulate a new sufficient condition for competitive equilibrium when there are only two types of products. Generalizations of our theorem in higher dimensions are equivalent to various forms of the Oda conjecture in algebraic geometry.

Gallagherian Prime Geodesic Theorem in Higher Dimensions

Using the Gallagher–Koyama approach, we reduce the exponent in the error term of the prime geodesic theorem for real hyperbolic manifolds with cusps.

Semiclassical Estimates for Scattering on the Real Line

We prove explicit semiclassical resolvent estimates for an integrable potential on the real line. The proof is a comparatively easy case of the spherical energies method, which has been used to prove similar theorems in higher dimensions and in more complicated geometric situations. The novelty in our results lies in the weakness of the assumptions on the potential.

Loop corrected soft photon theorem as a Ward identity

Recently Sahoo and Sen obtained a series of remarkable results concerning sub­ leading soft photon and graviton theorems in four dimensions. Even though the S-matrix is infrared divergent, they have shown that the subleading soft theorems are well defined and exact statements in QED and perturbative Quantum Gravity. However unlike the well studied Cachazo-Strominger soft theorems in tree-level amplitudes, the new subleading soft expansion is at the order ln w (where w is the soft frequency) and the corresponding soft factors structurally show completely different properties then their tree-level counterparts. Whence it is natural to ask if these theorems are associated to asymptotic symmetries of the S-matrix.We consider this question in the context of sub-leading soft photon theorem in scalar QED and show that there are indeed an infinity of conservation laws whose Ward identities are equivalent to the loop-corrected soft photon theorem. This shows that in the case of four dimensional QED, the leading and sub-leading soft photon theorems are equivalent to Ward identities of (asymptotic) charges.

A simplification problem in manifold theory

Two smooth manifolds M and N are called \mathbb R -diffeomorphic if M \times \mathbb R is diffeomorphic to N \times \mathbb R . We consider the following simplication problem: does \mathbb R -diffeomorphism imply diffeomorphism or homeomorphism? For compact manifolds, analysis of this problem relies on some of the main achievements of the theory of manifolds, in particular the h- and s-cobordism theorems in high dimensions and the spectacular more recent classication results in dimensions 3 and 4. This paper presents what is currently known about the subject as well as some new results about classications of \mathbb R -diffeomorphisms.

Slice theorem and orbit type stratification in infinite dimensions

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash–Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification.

Fatou and brothers Riesz theorems in the infinite-dimensional polydisc

We study the boundary behavior of functions in the Hardy spaces on the infinite-dimensional polydisc. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz- Zygmund type theorems for radial convergence of functions with Fourier spectrum supported on $$\mathbb{N}_0^\infty\cup(-\mathbb{N}_0^\infty)$$ . As a consequence one obtains easy new proofs of the brothers F. and M. Riesz Theorems in infinite dimensions, as well as being able to extend a result of Rudin concerning which functions are equal to the modulus of an H1 function almost everywhere to $$\mathbb{T}^\infty$$ . Finally, we provide counterexamples showing that the pointwise Fatou theorem is not true in infinite dimensions without restrictions to the mode of radial convergence even for bounded analytic functions.

Non-wandering Fatou Components for Strongly Attracting Polynomial Skew Products

We show a partial generalization of Sullivan's non-wandering domain theorem in complex dimension two. More precisely, we show the non-existence of wandering Fatou components for polynomial skew products of $ \mathbb{C}^2$ with an invariant attracting fiber, under the assumption that the multiplier $ \lambda $ is small. We actually show a stronger result, namely that every forward orbit of any vertical Fatou disk intersects a bulging Fatou component.

Marstrand's Theorem revisited: Projecting sets of dimension zero

We establish a refinement of Marstrand's projection theorem for Hausdorff dimension functions finer than the usual power functions, including an analogue of Marstrand's Theorem for logarithmic Hausdorff dimension.

Gravity waves from soft theorem in general dimensions

Classical limit of multiple soft graviton theorem can be used to compute the angular power spectrum of long wavelength gravitational radiation in classical scattering provided the total energy carried away by the radiation is small compared to the energies of the scatterers. We could ensure this either by taking the limit in which the impact parameter is large compared to the Schwarzschild radii of the scatterers, or by taking the probe limit where one object (the probe) has mass much smaller than the other object (the scatterer). We compute the results to subsubleading order in soft momentum and test them using explicit examples involving classical scattering. Our analysis also generalizes to the case where there are multiple objects involved in the scattering and the objects exchange mass, fragment or fuse into each other during the scattering. A similar analysis can be carried out for soft photons to subleading order, reproducing standard textbook results. We also discuss the modification of soft expansion in four dimensions beyond the leading order due to infrared divergences.