Finite Space Research Articles

A unified optimization design method for multi-stage non-circular gear transmission based on periodic B-spline interpolation

This research introduces a novel and unified optimization design method for multi-stage non-circular gear transmission (MNCGT) to address the challenges in designing MNCGT for complex motion requirements. The method optimizes non-circular gears comprehensively, reducing design and manufacturing difficulties while ensuring the realization of specified transmission requirements. A unified parameterization method, grounded on periodic B-spline interpolation, is introduced to establish the transmission function of non-circular gears and map it to a finite unified variable space. This innovative approach effectively reduces the difficulty and constraints of MNCGT optimization design. The proposed method takes into account crucial factors such as non-circularity, smoothness, processing conditions, and contact ratio, which significantly impact the transmission performance and manufacturing feasibility of non-circular gears. The effectiveness and superiority of this method are demonstrated through two practical examples and a real-world application in a planetary gear transplant mechanism, highlighting its potential for solving complex engineering problems.

Vector valued piecewise continuous almost automorphic functions and some consequences

In the present work, for X a Banach space, the notion of piecewise continuous Z-almost automorphic functions with values in finite dimensional spaces is extended to piecewise continuous Z-almost automorphic functions with values in X. Several properties of this class of functions are provided, in particular it is shown that if X is a Banach algebra, then this class of functions constitute also a Banach algebra; furthermore, using the theory of Z-almost automorphic functions, a new characterization of compact almost automorphic functions is given. As consequences, with the help of Z-almost automorphic functions, it is presented a simple proof of the characterization of almost automorphic sequences by compact almost automorphic functions; the method permits us to give explicit examples of compact almost automorphic functions which are not almost periodic. Also, using the theory developed here, it is shown that almost automorphic solutions of differential equations with piecewise constant argument are in fact compact almost automorphic. Finally, it is proved that the classical solution of the 1D heat equation with continuous Z-almost automorphic source is also continuous Z-almost automorphic; furthermore, we comment applications to the existence and uniqueness of the asymptotically continuous Z-almost automorphic mild solution to abstract integro-differential equations with nonlocal initial conditions.

A Scaling Relation, Zm-Type Deconfinement Phases, and Imaginary Chemical Potentials in Finite Temperature Large-N Gauge Theories

Abstract We show that the effective potentials for the Polyakov loops in finite temperature SU$(N)$ gauge theories obey a certain scaling relation with respect to temperature in the large-N limit. This scaling relation strongly constrains the possible terms in the Polyakov loop effective potentials. Moreover, by using the effective potentials in the presence of imaginary chemical potentials or imaginary angular velocities in several models, we find that phase transitions to $Z_m$-type deconfinement phases ($Z_m$ phase) occur, where the eigenvalues of the Polyakov loop are distributed $Z_m$ symmetrically. Physical quantities in the $Z_m$ phase obey the scaling properties of the effective potential. The models include Yang–Mills (YM) theories, the bosonic BFSS matrix model, and ${\mathcal {N}}=4$ supersymmetric YM theory on $S^3$. Thus, the phase diagrams of large-N gauge theories with imaginary chemical potentials are very rich and the stable $Z_m$ phase would be ubiquitous. Monte-Carlo calculations also support this. As a related topic, we discuss the phase diagrams of large-N YM theories with real angular velocities in finite volume spaces.

A Two-Phase Infinite/Finite Low-Level Memory Model: Reconciling Integer–Pointer Casts, Finite Space, and undef at the LLVM IR Level of Abstraction

A Two-Phase Infinite/Finite Low-Level Memory Model: Reconciling Integer–Pointer Casts, Finite Space, and undef at the LLVM IR Level of Abstraction

Descent modulus and applications

The norm of the gradient ‖∇f(x)‖ measures the maximum descent of a real-valued smooth function f at x. For (nonsmooth) convex functions, this is expressed by the distance dist(0,∂f(x)) of the subdifferential to the origin, while for general real-valued functions defined on metric spaces by the notion of metric slope |∇f|(x). In this work we propose an axiomatic definition of descent modulus T[f](x) of a real-valued function f at every point x, defined on a general (not necessarily metric) space. The definition encompasses all above instances as well as average descents for functions defined on probability spaces. We show that a large class of functions are completely determined by their descent modulus and corresponding critical values. This result is already surprising in the smooth case: a one-dimensional information (norm of the gradient) turns out to be almost as powerful as the knowledge of the full gradient mapping. In the nonsmooth case, the key element for this determination result is the break of symmetry induced by a downhill orientation, in the spirit of the definition of the metric slope. The particular case of functions defined on finite spaces is studied in the last section. In this case, we obtain an explicit classification of descent operators that are, in some sense, typical.

(Almost isometric) local retracts in metric spaces

We introduce the notion of (almost isometric) local retracts in metric space as a natural non-linear version of the concepts of ideals and almost isometric ideals in Banach spaces. We prove that given two metric spaces N⊆M there always exists an almost isometric local retract S⊆M with N⊆S and dens(N)=dens(S). We also prove that metric spaces which are local retracts (respectively almost isometric local retracts) can be characterised in terms of a condition of extendability of Lipschitz functions (respectively almost isometries) between finite metric spaces. Various examples and counterexamples are exhibited.

Park Space, Movement and Equity: Support of Physical Activity per Square Foot by Park Features.

Given the finite space available for parks in most urban areas, understanding the impact of design and park amenities on park visitation and physical activity should be considered when remodeling or creating new parks. This study analyzed park use and engagement in moderate-to-vigorous physical activity (MVPA) in specific park amenities across 198 parks in 27 US cities from the 2016 National Study of Neighborhood Parks based on each feature's square footage. The study also specifically measured use of park space by age group and gender. After mapping the parks, measuring the square feet of the most common amenities and controlling for factors like population density, neighborhood poverty levels, and park size, we found varied and inequitable use of amenities by age and gender, with men and boys having considerably greater use than women and girls. The findings suggest that park management and design should support more efficient, equitable, and beneficial use of public spaces.

Nontrivial t-designs in polar spaces exist for all t

AbstractA finite classical polar space of rank n consists of the totally isotropic subspaces of a finite vector space over $$\mathbb {F}_q$$ F q equipped with a nondegenerate form such that n is the maximal dimension of such a subspace. A t-$$(n,k,\lambda )$$ ( n , k , λ ) design in a finite classical polar space of rank n is a collection Y of totally isotropic k-spaces such that each totally isotropic t-space is contained in exactly $$\lambda $$ λ members of Y. Nontrivial examples are currently only known for $$t\le 2$$ t ≤ 2 . We show that t-$$(n,k,\lambda )$$ ( n , k , λ ) designs in polar spaces exist for all t and q provided that $$k>\frac{21}{2}t$$ k > 21 2 t and n is sufficiently large enough. The proof is based on a probabilistic method by Kuperberg, Lovett, and Peled, and it is thus nonconstructive.

Nikishin’s theorem and factorization through Marcinkiewicz spaces

Consider L 0 L^0 , the F F -space of all equivalence classes of measurable functions on a finite measure space equipped with the topology of convergence in measure. Inspired by Nikishin’s classical result on the factorization of sublinear continuous operators from a Banach space to L 0 L^0 , we prove a theorem that characterizes those maps from any quasi-metric space into L 0 L^0 that factor strongly through Marcinkiewicz weighted spaces. We show applications to sublinear operators on a certain class of quasi-Banach spaces with generalized Rademacher type generated by Orlicz sequence spaces.

On the optimal rate for the convergence problem in mean field control

The goal of this work is to obtain (nearly) optimal rates for the convergence problem in mean field control. Our analysis covers cases where the solutions to the limiting problem may not be unique nor stable. Equivalently the value function of the limiting problem might not be differentiable on the entire space. Our main result is then to derive sharp rates of convergence in two distinct regimes. First, when the data is sufficiently regular, we obtain rates proportional to N−1/2, with N being the number of particles, and we verify that N−1/2 is indeed optimal in this setting. Second, when the data is merely Lipschitz and semi-concave with respect to the first Wasserstein distance, we obtain rates proportional to N−2/(3d+6). We do not expect this second estimate to be optimal, but it improves substantially on the existing literature. Moreover, we construct an example showing that the optimal rate is no faster than N−1/d, and we conjecture that the optimal rate should indeed be exactly N−1/d (at least when d≥3). The key argument in our approach consists in mollifying the value function of the limiting problem in order to produce functions that are almost classical sub-solutions to the limiting Hamilton-Jacobi equation (which is a PDE set on the space of probability measures). These sub-solutions can be projected onto finite dimensional spaces and then compared with the value functions associated with the particle systems. In the end, this comparison is used to prove the most demanding bound in the estimates. The key challenge therein is thus to exhibit an appropriate form of mollification. We do so by employing sup-convolution within a convenient functional Hilbert space. To make the whole easier, we limit ourselves to the periodic setting. We also provide some examples to show that our results are sharp up to some extent.

On parallelisms of PG(3,4) with automorphisms of order 2

Let PG(n,q) be the n-dimensional projective space over the finite field Fq. A spread in PG(n,q) is a set of mutually skew lines which partition the point set. A parallelism is a partition of the set of lines by spreads. The classification of parallelisms in small finite projective spaces is of interest for problems from projective geometry, design theory, network coding, error-correcting codes, and cryptography. All parallelisms of PG(3,2) and PG(3,3) are known. Parallelisms of PG(3,4) which are invariant under automorphisms of odd prime orders have also been classified. The present paper contributes to the classification of parallelisms in PG(3,4) with automorphisms of even order. We focus on cyclic groups of order four and the group of order two generated by a Baer involution. We examine invariants of the parallelisms such as the full automorphism group, the type of spreads and questions of duality. The results given in this paper show that the number of parallelisms of PG(3,4) is at least 8675365. Some future directions of research are outlined.

Exact lattice chiral symmetry in 2D gauge theory

We construct symmetry-preserving lattice regularizations of 2D QED with one and two flavors of Dirac fermions, as well as the “3450” chiral gauge theory, by leveraging bosonization and recently proposed modifications of Villain-type lattice actions. The internal global symmetries act just as locally on the lattice as they do in the continuum, the anomalies are reproduced at finite lattice spacing, and in each case we find a sign-problem-free dual formulation. Published by the American Physical Society 2024

On the asymptotic behaviour of the fractional Sobolev seminorms: A geometric approach

We study the well-known asymptotic formulas for fractional Sobolev functions à la Bourgain–Brezis–Mironescu and Maz'ya–Shaposhnikova, in a geometric approach. We show that the key to these asymptotic formulas are Rademacher's theorem and volume growth at infinity respectively. Examples fitting our framework includes Euclidean spaces, Riemannian manifolds, Alexandrov spaces, finite dimensional Banach spaces, and some ideal sub-Riemannian manifolds.

Finite element approximation of unique continuation of functions with finite dimensional trace

We consider a unique continuation problem where the Dirichlet trace of the solution is known to have finite dimension. We prove Lipschitz stability of the unique continuation problem and design a finite element method that exploits the finite dimensionality to enhance stability. Optimal a priori and a posteriori error estimates are shown for the method. The extension to problems where the trace is not in a finite dimensional space, but can be approximated to high accuracy using finite dimensional functions is discussed. Finally, the theory is illustrated in some numerical examples.

Recurrence formula for some higher order evolution equations

Riccati’s differential equation is formulated as abstract equation in finite or infinite dimensional Banach spaces. Since the Riccati’s differential equation with the Cole–Hopf transform shows a relation between the first order evolution equations and the second order evolution equations, its generalization suggests the existence of recurrence formula leading to a sequence of differential equations with different order. In conclusion, by means of the logarithmic representation of operators, a transform between the first order evolution equations and the higher order evolution equation is presented. Several classes of evolution equations with different orders are given, and some of them are shown as examples.

Generalized Frames in Finite Space and Applications

In the field of signal and imaging processing, the Frame Theory is a practical-mathematical tool for analysis. Moreover, generalized frames such as K-frame and g-frame are also hot research topics in the field of analytics. We apply the conclusions of the frames in infinite-dimensional to the finite-dimensional space, which makes it easier for us to apply the conclusions in actual scenarios.

Semigroups of linear transformations whose restrictions belong to a general linear group

Let V be a vector space and U a fixed subspace of V. We denote the semigroup of all linear transformations on V under composition of functions by L(V). In this paper, we study the semigroup of all linear transformations on V whose restrictions belong to the general linear group GL(U), denoted by L GL ( U ) ( V ) . More precisely, we consider the subsemigroup L GL ( U ) ( V ) = { α ∈ L ( V ) : α | U ∈ GL ( U ) } of L(V). In this work, Green’s relations and ideals of this semigroup are described. Then we also determine the minimal ideal and the set of all minimal idempotents of it. Moreover, we establish an isomorphism theorem when V is a finite dimensional vector space over a finite field. Finally, we find its generating set.

Fixed-Point Theorems for Fuzzy Mappings

Since the 1970s and 1980s, significant contributions have been made by Weiss, Butnariu, Heilpern, Chitra, Subrahmanyam, and others, extending fixed-point theorems to fuzzy mappings and topological spaces. This paper provides two generalizations of two important fixed-point theorems, one provided by Butnariu and the other provided by Chitra. The first generalization ensures that, under certain conditions, an acyclic fuzzy mapping has a fixed point. The second result ensures the existence of a point in the intersection of two or more fuzzy mappings considering contractible finite dimensional ANR spaces, which is a generalization of the statement provided by Chitra.

Continuum-extrapolated high-order baryon fluctuations

Fluctuations play a key role in the study of QCD phases. Lattice QCD is a valuable tool to calculate them, but going to high orders is challenging. Up to the fourth order, continuum results have been available since 2015. We present the first continuum results for sixth-order baryon fluctuations for temperatures between T=130 and 200 MeV and eighth order at T=145 MeV in a fixed volume. Comparison with earlier studies with imaginary chemical potential suggests that the volume effect is under control for T<145 MeV. Our results are in sharp contrast with well-known results in the literature obtained at finite lattice spacing. Published by the American Physical Society 2024

On Finite Topological Spaces Generated by Connected Simple Graphs

In this paper, we investigate topologies produced by simple connected graphs. In particular, we will present the type of relations on the vertices set. It is converted to an adjacent matrix and through this matrix, whose elements represent the relationship between each vertex to the rest of the vertices adjacent to it, where sets are produced through which they represent the basis for topology. On the graph is the vertex set.