The X = M conjecture of Hatayama et al. asserts the equality between the one-dimensional configuration sum X expressed as the generating function of crystal paths with energy statistics and the fermionic formula M for all affine Kac–Moody algebra. In this paper we prove the X = M conjecture for tensor products of Kirillov–Reshetikhin crystals B1,s associated to symmetric powers for all nonexcep...

The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial semantics from final semantics, using the initiality and finality to ensure their equality. Moreo...

We present a short elementary proof of the Lagrange multiplier theorem for equality-constrained optimization. Most proofs in the literature rely on advanced analysis concepts such as the implicit function theorem, whereas elementary proofs tend to be long and involved. By contrast, our proof uses only basic facts from linear algebra, the definition of differentiability, the critical-point condi...

We use the notation A ∈ F to indicate that the set A is in F . Remark that since A∩B = (A ∪B) and more generally ⋂ Ai = ( ⋃ Ai) c (Exercise: Prove this equality) axiom ii) holds with union replaced by intersection. Also we have A−B = A ∩B so if A,B ∈ F then A−B ∈ F . Basically you should think of a σ-algebra as a system of subsets in which you can perform any of the usual set-theoretic operatio...

We construct a realization of the algebra of the Z3-graded topological symmetry of type (1, 1, 1) in terms of a pair of operators D1 : H1 → H2, D2 : H2 → H3 satisfying [D1D † 1,D † 2D2] = 0. We show that the sequence of the restriction of these operators to the zero-energy subspace forms a complex and establish the equality of the corresponding topological invariants with the analytic indices o...

The X = M conjecture of Hatayama et al. asserts the equality between the one-dimensional configuration sum X expressed as the generating function of crystal paths with energy statistics and the fermionic formula M for all affine Kac–Moody algebra. In this paper we prove the X = M conjecture for tensor products of Kirillov–Reshetikhin crystals B1,s associated to symmetric powers for all nonexcep...

We study fragments of Allen’s algebra that contain a basic relation distinct from the equality relation. We prove that such a fragment is either NPcomplete or else contained in some already known tractable subalgebra. We obtain this result by giving a new uniform description of known maximal tractable subalgebras and then systematically using an algebraic technique for description of maximal su...

We consider the C*-algebraRq generated by the representation of the q-commutation relations on the twisted Fock space. We construct a canonical unitary U (= U(q)) from the twisted Fock space to the usual Fock space, such that URqU∗ contains the extended Cuntz algebra R0, for all q ∈ (−1, 1). We prove the equality URqU∗ = R0 for q satisfying: q < 1− 2|q|+ 2|q| − 2|q| + · · ·+ 2(−1)k|q|k2 + · · ·...

We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally close...

We prove that the property of excision in algebraic K-theory is for a Q-algebra A equivalent to the H-unitality of the latter. Our excision theorem, in particular, implies Karoubi's conjecture on the equality of algebraic and topological K-theory groups of stable C*-algebras. It also allows us to identify the algebraic K-theory of the symbol map in the theory of pseudodifferential operators.