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 Ngai-Ching Wong Mathematics , 2005, Abstract: This paper discusses two common techniques in functional analysis: the topological method and the bornological method. In terms of Pietsch's operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of Grothendieck (via Banach space operators), Randtke (via continuous seminorms) and Hogbe-Nlend (via convex bounded sets) are compared.
 Mathematics , 2014, Abstract: Rough set theory is an important mathematical tool for dealing with uncertain or vague information. This paper studies some new topologies induced by a binary relation on universe with respect to neighborhood opera- tors. Moreover, the relations among them are studied. In additionally, lower and upper approximations of rough sets using the binary relation with respect to neighborhood operators are studied and examples are given.
 Mathematics , 2010, Abstract: We consider the locally measure topology $t(\mathcal{M})$ on the *-algebra $LS(\mathcal{M})$ of all locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$. We prove that $t(\mathcal{M})$ coincides with the $(o)$-topology on $LS_h(\mathcal{M})=\{T\in LS(\mathcal{M}): T^*=T\}$ if and only if the algebra $\mathcal{M}$ is $\sigma$-finite and a finite algebra. We study relationships between the topology $t(\mathcal{M})$ and various topologies generated by faithful normal semifinite traces on $\mathcal{M}$.
 C. B. Lang Physics , 1999, Abstract: With the Schwinger model as example I discuss properties of lattice Dirac operators, with some emphasis on Monte Carlo results for topological charge, chiral fermions and eigenvalue spectra.
 Martha Lee Hollist Kilpack Mathematics , 2014, Abstract: On an infinite set some closure operators are finitary (algebraic) while others are not. We can generalize this idea for a complete algebraic lattice letting the compact elements act as the finite sets. With this in mind, we will consider the set of algebraic closure operators on such a lattice. We will show this set forms a complete lattice that is also an algebraic lattice.
 Panagiotis D. Stratigos International Journal of Mathematics and Mathematical Sciences , 1992, DOI: 10.1155/s0161171292000905 Abstract: We consider the general setting of A.D. Alexandroff, namely, an arbitrary set X and an arbitrary lattice of subsets of X, ￠ ’. ° ’ ( ￠ ’) denotes the algebra of subsets of X generated by ￠ ’ and MR( ￠ ’) the set of all lattice regular, (finitely additive) measures on ° ’ ( ￠ ’).
 Werner Kerler Physics , 2002, DOI: 10.1142/S0217751X03013910 Abstract: We only require generalized chiral symmetry and $\gamma_5$-hermiticity, which leads to a large class of Dirac operators describing massless fermions on the lattice, and use this framework to give an overview of developments in this field. Spectral representations turn out to be a powerful tool for obtaining detailed properties of the operators and a general construction of them. A basic unitary operator is seen to play a central r\^ole in this context. We discuss a number of special cases of the operators and elaborate on various aspects of index relations. We also show that our weaker conditions lead still properly to Weyl fermions and to chiral gauge theories.
 Physics , 2009, Abstract: We prove that the interval topology of an Archimedean atomic lattice effect algebra $E$ is Hausdorff whenever the set of all atoms of $E$ is almost orthogonal. In such a case $E$ is order continuous. If moreover $E$ is complete then order convergence of nets of elements of $E$ is topological and hence it coincides with convergence in the order topology and this topology is compact Hausdorff compatible with a uniformity induced by a separating function family on $E$ corresponding to compact and cocompact elements. For block-finite Archimedean atomic lattice effect algebras the equivalence of almost orthogonality and s-compact generation is shown. As the main application we obtain the state smearing theorem for these effect algebras, as well as the continuity of $\oplus$-operation in the order and interval topologies on them.
 Brian Jefferies Journal of Operators , 2014, DOI: 10.1155/2014/629502 Abstract: A bounded linear operator on a Hilbert space is trace class if its singular values are summable. The trace class operators on form an operator ideal and in the case that is finite-dimensional, the trace tr of is given by for any matrix representation of . In applications of trace class operators to scattering theory and representation theory, the subject is complicated by the fact that if is an integral kernel of the operator on the Hilbert space with a -finite measure, then may not be defined, because the diagonal may be a set of -measure zero. The present note describes a class of linear operators acting on a Banach function space which forms a lattice ideal of operators on , rather than an operator ideal, but coincides with the collection of hermitian positive trace class operators in the case of . 1. Introduction A trace class operator on a separable Hilbert space is a compact operator whose singular values , , satisfy The decreasing sequence consists of eigenvalues of . Equivalently, is trace class if and only if, for any orthonormal basis of , the sum is finite. The number is called the trace of and is independent of the orthonormal basis of . Lidskii’s equality asserts that is actually the sum of the eigenvalues of the compact operator [1, Theorem 3.7]. We refer to [1] for properties of trace class operators. The collection of trace class operators on is an operator ideal and Banach space with the norm . The following facts are worth noting in the case of the Hilbert space with respect to Lebesgue measure on the interval .(a)If is a trace class linear operator, then there exist , , with and ？where a.e.. In particular, is regular and has an integral kernel . Moreover, (b)Suppose that is a regular linear operator defined by formula (3) for a continuous function . If is trace class, then , and [2, Theorem ].(c)Suppose that the function is continuous and positive definite; that is, for all and , , and any . Then for all . If , then there exists a unique trace class operator defined by formula (3) [1, Theorem 2.12]. Let be a measure space. The projective tensor product is the set of all sums: The norm of is given by where the infimum is taken over all sums for which the representation (5) holds. The Banach space is actually the completion of the algebraic tensor product with respect to the projective tensor product norm [3, Section 6.1]. There is a one-to-one correspondence between the space of trace class operators acting on and , so that the trace class operator has an integral kernel . If the integral kernel given by (5) has the property that for
 Physics , 2011, DOI: 10.1103/PhysRevD.85.014507 Abstract: Motivated by the desire to construct meson-meson operators of definite relative momentum in order to study resonances in lattice QCD, we present a set of single-meson interpolating fields at non-zero momentum that respect the reduced symmetry of a cubic lattice in a finite cubic volume. These operators follow from the subduction of operators of definite helicity into irreducible representations of the appropriate little groups. We show their effectiveness in explicit computations where we find that the spectrum of states interpolated by these operators is close to diagonal in helicity, admitting a description in terms of single-meson states of identified J^{PC}. The variationally determined optimal superpositions of the operators for each state give rapid relaxation in Euclidean time to that state, ideal for the construction of meson-meson operators and for the evaluation of matrix elements at finite momentum.
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