Abstract. In this article, we study the Berglund–Hübsch transpose construction W T for invertible quasihomogeneous potential W . We introduce the dual group G and establish the state space isomorphism between the Fan–Jarvis–Ruan–Witten A-model of W/G and the orbifold Milnor ring B-model of W T /G . Furthermore, we prove a mirror symmetry theorem at the level of Frobenius algebra structure for G...

• The dual problem is always convex no matter if the primal problem is convex, i.e., g is always concave. • The primal and dual optimal values, f∗ and g∗, always satisfy weak duality: f∗ ≥ g∗. • Slater’s condition: for convex primal, if there is an x such that h1(x) < 0, · · · , hm(x) < 0 and l1(x) = 0, · · · , lr(x) = 0 (12.5) then strong duality holds: f∗ = g∗. Note that the condition can be ...

Let G be a torsion free discrete group with a finite dimensional classifying space BG. We show that G has a dual Dirac morphism if and only if a certain coarse (co)-assembly map is an isomorphism. Hence the existence of a dual Dirac morphism for such G is a metric, that is, coarse, invariant of G. We get similar results for groups with torsion as well. The framework that we develop is also suit...

In this paper, we investigate duality of modular g-Riesz bases and g-Riesz bases in Hilbert C*-modules. First we give some characterization of g-Riesz bases in Hilbert C*modules, by using properties of operator theory. Next, we characterize the duals of a given g-Riesz basis in Hilbert C*-module. In addition, we obtain sufficient and necessary condition for a dual of a g-Riesz basis to be again...

The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which receive distinct colours. For a connected cubic plane graph G we prove that n 2 + ∗1 − 1 rb(G) n− ∗0 + 1, where ∗0 and ∗1 denote the independence number and the edge independence number, respectively, of th...

For G a locally compact group and G its dual, let Jd(G) be the C*algebra generated by the Fourier-Stieltjes transforms of the discrete measures on G. We show that the canonical trace on ,4(G) is faithful if and only if G is amenable as a discrete group. We further show that if G is nondiscrete and amenable as a discrete group, then the only measures in dKd(G) are the discrete measures, and also...

Write S = {z ∈ C : |z| = 1}. A character of a locally compact abelian group G is a continuous group homomorphism G → S. We denote by Ĝ the set of characters of G, where for φ1, φ2 ∈ Ĝ and x ∈ G, we define (φ1φ2)(x) = φ1(x)φ2(x). We assign Ĝ the final topology for the family of functions {φ 7→ φ(x) : x ∈ G}, i.e., the coarsest topology on Ĝ so that for each x ∈ G, the function φ 7→ φ(x) is conti...

The Mishchenko–Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra g there exists a complete set of commuting polynomials on its dual space g*. In terms of the theory of integrable Hamiltonian systems this means that the dual space g* endowed with the standard Lie–Poisson bracket admits polynomial integrable Hamiltonian systems. This conjecture was proved by S. ...

A graph parameter is self-dual in some class of graphs embeddable in some surface if its value does not change in the dual graph more than a constant factor. Self-duality has been examined for several width-parameters, such as branchwidth, pathwidth, and treewidth. In this paper, we give a direct proof of the self-duality of branchwidth in graphs embedded in some surface. In this direction, we ...

Let G be a 3-connected planar graph and G∗ be its dual. We show that the pathwidth of G∗ is at most 6 times the pathwidth of G. We prove this result by relating the pathwidth of a graph with the cut-width of its medial graph and we extend it to bounded genus embeddings. We also show that there exist 3-connected planar graphs such that the pathwidth of such a graph is at least 1.5 times the path...