Borel Equivalence Relations between l_1and l_p

In this paper, we show that, for each p 〉 1, there are continuum many Borel equivalence relations between Rω/l1 and Rω/p ordered by ≤B which are pairwise Borel incomparable.

Einstein’s Dark Energy via Similarity Equivalence, ‘tHooft Dimensional Regularization and Lie Symmetry Groups

Realizing the physical reality of ‘tHooft’s self similar and dimensionaly regularized fractal-like spacetime as well as being inspired by a note worthy anecdote involving the great mathematician of Alexandria, Pythagoras and the larger than life man of theoretical physics Einstein, we utilize some deep mathematical connections between equivalence classes of equivalence relations and E-infinity theory quotient space. We started from the basic principles of self similarity which came to prominence in science with the advent of the modern theory of nonlinear dynamical systems, deterministic chaos and fractals. This fundamental logico-mathematical thread related to partially ordered sets is then applied to show how the classical Newton’s kinetic energy E = 1/2mv2 leads to Einstein’s celebrated maximal energy equation E = mc2 and how in turn this can be dissected into the ordinary energy density E(O) = mc2/22 and the dark energy density E(D) = mc2(21/22) of the cosmos where m is the mass;v is the velocity and c is the speed of light. The important role of the exceptional Lie symmetry groups and ‘tHooft-Veltman-Wilson dimensional regularization in fractal spacetime played in the above is also highlighted. The author hopes that the unusual character of the analysis and presentation of the present work may be taken in a positive vein as seriously attempting to propose a different and new way of doing theoretical physics by treating number theory, set theory, group theory, experimental physics as well as conventional theoretical physics on the same footing and letting all these diverse tools lead us to the answer of fundamental questions without fear of being labelled in one way or another.

The Equivalence between Orthogonal Iterations and Alternating Least Squares

This note explores the relations between two different methods. The first one is the Alternating Least Squares (ALS) method for calculating a rank-k approximation of a real m×n matrix, A. This method has important applications in nonnegative matrix factorizations, in matrix completion problems, and in tensor approximations. The second method is called Orthogonal Iterations. Other names of this method are Subspace Iterations, Simultaneous Iterations, and block-Power method. Given a real symmetric matrix, G, this method computes k dominant eigenvectors of G. To see the relation between these methods we assume that G = AT A. It is shown that in this case the two methods generate the same sequence of subspaces, and the same sequence of low-rank approximations. This equivalence provides new insight into the convergence properties of both methods.

Generalized Szasz Problems in Normal Classes of Algebras

Let A be a normal class of algebras. In the present paper, we characterize the following four problems for A: for which radical class R, there holds that(1) R(i1∧i2) = R(i1)∧R(i2);(2) R(i1∨i2) = R(i1)∨R(i2);(3) (i1∧i2) =i1∧i2;(4) (i1∨i2) = i1∨i2, for arbitrary algebra a∈A and any i1,i2∈La,where j denotes the ideal of a uniquely determined by R(a/j) = j/j?

The statistical laws of strong seismic motion phase characteristics

Under the assumption of an effective point source model,this paper introduces a conception of equivalent phase velocity to describle the average propagation processes and dispersion properties of seismic waves from seismic source to the site,then illuminates the relation between the phase properties of seismic motion and equivalent phase velocity and obtains a simple formula of estimating the equivalent phase velocity through the use of the phase difference spectrum of seismic motion.After the parts of strong seismic records in the western American and the SMART-1 array are used to identify the method presented by this paper in reason,the statistical relations of equivalent phase velocity are given in the above two regions.The results of the paper have demonstrated that the phase spectrum of seismic motion has its inner law as same as the amplitude spectrum does.

On C^(*)-algebras from homoclinic equivalences on subshifts of finite type

Let G be an infinite countable group and A be a finite set.IfΣ?A~G is a strongly irreducible subshift of finite type,we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation■onΣand show that the reduced C^(*)-algebra C_(r)^(*)(■)of■is a unital simple approximately finite(AF)-dimensional C^(*)-algebra.The shift action G of onΣinduces a canonical automorphism action of G on the C^(*)-algebra C_(r)^(*)(■).We give the notion of noncommutative dynamical entropy invariants for amenable group actions on C^(*)-algebras,and show that,if G is an amenable group,then the noncommutative topological entropy of the canonical automorphism action of G on C_(r)^(*)(■)is equal to the topology entropy of the shift action of G onΣ.We also establish the variational principle with respect to the noncommutative measure entropy and the topological entropy for the C^(*)-dynamical system(C_(r)^(*)(■),G).

Relative Derived Equivalences and Relative Homological Dimensions

Let A be a small abelian category.For a closed subbifunctor F of Ext_A^1（-,-）,Buan has generalized the construction of Verdier’s quotient category to get a relative derived category,where he localized with respect to F-acyclic complexes.In this paper,the homological properties of relative derived categories are discussed,and the relation with derived categories is given.For Artin algebras,using relative derived categories,we give a relative version on derived equivalences induced by F-tilting complexes.We discuss the relationships between relative homological dimensions and relative derived equivalences.

Hierarchical Covering Algorithm

The concept of deep learning has been applied to many domains, but the definition of a suitable problem depth has not been sufficiently explored. In this study, we propose a new Hierarchical Covering Algorithm （HCA） method to determine the levels of a hierarchical structure based on the Covering Algorithm （CA）. The CA constructs neural networks based on samples＇ own characteristics, and can effectively handle multi-category classification and large-scale data. Further, we abstract characters based on the CA to automatically embody the feature of a deep structure. We apply CA to construct hidden nodes at the lower level, and define a fuzzy equivalence relation R on upper spaces to form a hierarchical architecture based on fuzzy quotient space theory. The covering tree naturally becomes from R. HCA experiments performed on MNIST dataset show that the covering tree embodies the deep architecture of the problem, and the effects of a deep structure are shown to be better than having a single level.