Narrow Results

We present some recurrence results in the context of ergodic theory and dynamical systems. The main focus will be on smooth dynamical systems, in particular, those with some chaotic/hyperbolic behavior. The aim is to compute recurrence rates, limiting distributions of return times, and short returns. We choose to give the full proofs of the results directly related to recurrence, avoiding as much as possible to hide the ideas behind technical details. This drove us to consider as our basic dynamical system a one-dimensional expanding map of the interval. We note, however, that most of the arguments still apply to higher dimensional or less uniform situations, so that most of the statements continue to hold. Some basic notions from the thermodynamic formalism and the dimension theory of dynamical systems will be recalled.

We consider a matrix space based on the spin degree of freedom, describing both a Hilbert state space and its corresponding symmetry operators. Under the requirement that the Lorentz symmetry be kept, at given dimension, scalar symmetries and their representations are determined. Symmetries are flavor or gauge-like, with fixed chirality. After spin 0, 1/2 and 1 fields are obtained in this space, we construct associated interactive gauge-invariant renormalizable terms, showing their equivalence to a Lagrangian formulation, using as example the previously studied (5+1)-dimensional case, with many standard-model connections. At 7+1 dimensions, a pair of Higgs-like scalar Lagrangian is obtained naturally producing mass hierarchy within a fermion flavor doublet.

We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to products of sets in finite groups, in particular to word maps, and a generalization of Tao's Algebraic Regularity Lemma is noted.

We investigate geometrical properties and inequalities satisfied by the complex difference body, in the sense of studying which of the classical ones for the difference body have an analog in the complex framework. Among others we give an equivalent expression for the support function of the complex difference body and prove that, unlike the classical case, the dimension of the complex difference body depends on the position of the body with respect to the complex structure of the vector space. We use spherical harmonics to characterize the bodies for which the complex difference body is a ball, we prove that it is a polytope if and only if the two bodies involved in the construction are polytopes and provide several inequalities for classical magnitudes of the complex difference body, as volume, quermassintegrals and diameter, in terms of the corresponding ones for the involved bodies.

The tensor network variety is a variety of tensors associated to a graph and a set of positive integer weights on its edges, called bond dimensions. We determine an upper bound on the dimension of the tensor network variety. A refined upper bound is given in cases relevant for applications such as varieties of matrix product states and projected entangled pairs states. We provide a range (the “supercritical range”) of the parameters where the upper bound is sharp.

For an equilibrium measure of a Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero.

A noticeable consequence is that when this measure is not absolutely continuous, the probability that a ball of radius ε chosen at random have a measure smaller (or larger) than ε^δ is asymptotically equal to 1/2, where δ is the Hausdorff dimension of the measure.

Our method applies to a class of non-conformal expanding maps on the d-dimensional torus. It also applies to conformal repellers and Axiom A surface diffeomorphisms and possibly to a class of one-dimensional non-uniformly expanding maps. These generalizations are presented at the end of the paper.

For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J(L) of join-irreducible elements of L and the join-dependency relation D_L on J(L). We establish a similar version of this result for the dimension monoid Dim L of L, a natural precursor of Con L. For L join-semidistributive, this result takes the following form:

Theorem 1. Let L be a finite join-semidistributive lattice. Then Dim L is isomorphic to the commutative monoid defined by generators Δ(p), for p ∈ J(L), and relations

As a consequence of this, we obtain the following results:

Theorem 2. Let L be a finite join-semidistributive lattice. Then L is a lower bounded homomorphic image of a free lattice iff Dim L is strongly separative, iff it satisfies the axiom

Theorem 3. Let A and B be finite join-semidistributive lattices. Then the box product A □ B of A and B is join-semidistributive, and the following isomorphism holds:

In this paper, we first give an alternative characterization of the derived functor Ext via the Quillen model structure on the category of complexes induced by a given cotorsion pair in the category of modules, then based on this, we consider homological dimensions of complexes related to . As applications, we extend Gorenstein projective dimension of homologically bounded below complexes (in the sense of Christensen and coauthors) to unbounded complexes whenever R is Gorenstein. Moreover, we extend Stenström's FP-injective dimension from modules to complexes, define FP-projective dimension for complexes, and characterize Noetherian and von Neumann regular rings by these dimensions.

We study problems related to indecomposability of modules over certain local finite-dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of graph of cyclic modules, of combinatorial dimension, and of fundamental combinatorial dimension of a module. We use these concepts to establish, under favorable conditions, criteria for the indecomposability of a module. We present categorified versions of these constructions and we use this categorical framework to establish criteria for the indecomposability of modules of infinite rank.

When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, $\nu$-semifield) rather than a field. One needs to utilize congruences rather than ideals, leading to a considerably more complicated theory. In his dissertation, the first author exploited the multiplicative structure of an idempotent semifield, which is a lattice ordered group, in place of the additive structure, in order to apply the extensive theory of chains of homomorphisms of groups. Reworking his dissertation, starting with a semifield${}^{†}$$F$, we pass to the semifield${}^{†}$$F({\lambda }_1,\dots ,{\lambda }_n)$ of fractions of the polynomial semiring${}^{†}$, for which there already exists a well developed theory of kernels, which are normal convex subgroups of $F({\lambda }_1,\dots ,{\lambda }_n)$; the parallel of the zero set now is the $1$-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to $\nu$-kernels (Definition 4.1.4) and $1^{\nu }$-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The $\nu$-kernels corresponding to tropical hypersurfaces are the $1^{\nu }$-sets of what we call “corner internal rational functions,” and we describe $\nu$-kernels corresponding to “usual” tropical geometry as $\nu$-kernels which are “corner-internal” and “regular.” This yields an explicit description of tropical affine varieties in terms of various classes of $\nu$-kernels. The literature contains many tropical versions of Hilbert’s celebrated Nullstellensatz, which lies at the foundation of algebraic geometry. The approach in this paper is via a correspondence between $1^{\nu }$-sets and a class of $\nu$-kernels of the rational $\nu$-semifield${}^{†}$ called polars, originating from the theory of lattice-ordered groups. When $F$ is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal$\nu$-kernels, intersected with the $\nu$-kernel generated by $F$. For our main application, we develop algebraic notions such as composition series and convexity degree, leading to a dimension theory which is catenary, and a tropical version of the Jordan–Hölder theorem for the relevant class of $\nu$-kernels.

For the bounded derived category of an abelian category, bounds of the dimension with respect to a complete hereditary cotorsion pair are given. We also characterize levels of DG-modules and study how levels involved in a recollement of derived categories over DG-rings are related.

In various areas of modern physics and in particular in quantum gravity or foundational space–time physics, it is of great importance to be in the possession of a systematic procedure by which a macroscopic or continuum limit can be constructed from a more primordial and basically discrete underlying substratum, which may behave in a quite erratic and irregular way. We develop such a framework within the category of general metric spaces by combining recent work of our own and ingeneous ideas of Gromov et al. developed in pure mathematics. A central role is played by two core concepts. For one, the notion of intrinsic scaling dimension of a (discrete) space or, in mathematical terms, the growth degree of a metric space at infinity, on the other hand, the concept of a metrical distance between general metric spaces and an appropriate scaling limit (called by us a geometric renormalization group) performed in this metric space of spaces. In doing this, we prove a variety of physically interesting results about the nature of this limit process, properties of the limit space, e.g., what preconditions qualify it as a smooth classical space–time and, in particular, its dimension.

In this paper, we study the class of complexes of FP-injective left R-modules. It is shown that the pair is a complete cotorsion pair. If R is a left coherent ring, it is proved that every complex has an -cover. We also introduce the FP-injective dimension of complexes. A special attention is paid to the dimension of homologically bounded above R-complexes over a left coherent ring which has a nice functorial description.

Let $T$ be a tilting module. In this paper, Gorenstein $\sigma [T]$-injective modules are introduced and some of their basic properties are studied. Moreover, some characterizations of rings over which all modules are Gorenstein $\sigma [T]$-injective are given. For instance, we show that every $R$-module is Gorenstein $\sigma [T]$-injective if and only if every flat $R$-module is Gorenstein $\sigma [T]$-injective if and only if $R$ is a $\sigma [T]$-injective module on itself.

Currently, many new materials including nanomaterials exhibit exotic properties like, mention a few, unconventional superconductivity, reduced magnetism, reduced Pauli repulsion. These exotic properties could not be explained completely with the existing quantum mechanical theories which did not take into account the dimension of angle. Now, there arises a necessity to invoke the forgotten dimension of angle, for explaining these exotic interesting properties. So far angle is considered as a dimensionless quantity even though its all time partner $r$ is endowed with dimension. In materials, where surface plays a dominant role, this new dimension, of angle becomes an important tool for exploring their morphology and exotic properties. This new dimension of angle restores the orbital degrees of freedom to the s electrons whose orbital degrees of freedom are suppressed ($\ell =0$). It compensates for the missed signature of s electron orbital angular momentum.

In this article we examine fractal curves and synthesis algorithms in musical composition and research. First we trace the evolution of different approaches for the use of fractals in music since the 80's by a literature review. Furthermore, we review representative fractal algorithms and platforms that implement them. Properties such as self-similarity (pink noise), correlation, memory (related to the notion of Brownian motion) or non correlation at multiple levels (white noise), can be used to develop hierarchy of criteria for analyzing different layers of musical structure. L-systems can be applied in the modelling of melody in different musical cultures as well as in the investigation of musical perception principles. Finally, we propose a critical investigation approach for the use of artificial or natural fractal curves in systematic musicology.

The symbolic representation of quantities, units, and dimensions is reviewed with regard to information carried by conventional notation. The formal structure of unit systems like the International System of Units (SI) has been an efficient way for scientists to share data about the magnitudes of quantities, but it is an unsatisfactory basis for digitalisation. SI notation does not convey information about kind of quantity, which must be obtained from some other source, and the use of dimensional expressions in the SI also loses information about kinds of quantity. It is argued that digitalisation of quantities, units, and dimensions should focus on the more fundamental measurement concepts of quantity kinds and scales of measurement to provide strong foundations on which to build support for quantities, including features familiar from SI unit notation.