[en] We show how a recently proposed large N duality in the context of type IIA strings with N=1 supersymmetry in 4 dimensions can be derived from purely geometric considerations by embedding type IIA strings in M-theory. The phase structure of M-theory on G₂ holonomy manifolds and an S³ flop are the key ingredients in this derivation

[en] The necessary and sufficient conditions are shown for the duality, twisted duality and weak twisted duality of observable algebra and field algebra

[en] Inspired by the multiplicative nature of the Ramanujan modular discriminant, Δ, we consider physical realizations of certain multiplicative products over the Dedekind eta-function in two parallel directions: the generating function of BPS states in certain heterotic orbifolds and elliptic K3 surfaces associated to congruence subgroups of the modular group. We show that they are, after string duality to type II, the same K3 surfaces admitting Nikulin automorphisms. In due course, we will present identities arising from q-expansions as well as relations to the sporadic Mathieu group M_2_4

[en] We propose the notion of quantum coherence for superpositions over states which are not necessarily mutually orthogonal. This anticipatedly leads to a resource theory of non-orthogonal coherence. We characterize free states and free operations in this theory, and connect the latter with free operations in the resource theory of quantum coherence for orthogonal bases. We show that the concept of non-orthogonal coherence naturally furnishes us with a wave-particle duality in quantum double-slit experiments where the channels beyond the slits are leaky between them. Furthermore, we demonstrate existence of a unique maximally coherent qubit state corresponding to any given purity. In addition, and in contradistinction with the case of orthogonal bases, there appears a non-trivial minimally coherent qubit state for a given purity. We also study the behavior of quantum coherence for some typical configurations of non-orthogonal bases which have no analogs for orthogonal bases. We further investigate the problem of determining the energy cost of creating non-orthogonal coherence, and find that it scales linearly with the non-orthogonal coherence created. (paper)

[en] Highlights: • Fractal-fractional differentiation. • Fractal-fractional integration. • New numerical scheme for fractal-fractional operators. • New model of Darcy scale describing flow in a dual medium. - Abstract: New operators of differentiation have been introduced in this paper as convolution of power law, exponential decay law, and generalized Mittag-Leffler law with fractal derivative. The new operators will be referred as fractal-fractional differential and integral operators. The new operators aimed to attract more non-local natural problems that display at the same time fractal behaviors. Some new properties are presented, the numerical approximation of these new operators are also presented with some applications to real world problem.

[en] We give a generalization of the theorem of Bondal and Orlov about the derived categories of coherent sheaves on intersections of quadrics, revealing the relation of this theorem to projective duality. As an application, we describe the derived categories of coherent sheaves on Fano 3-folds of index 1 and degrees 12, 16 and 18

[en] A new approach to convex calculus is presented, which allows one to treat from a single point of view duality and calculus for various convex objects. This approach is based on the possibility of associating with each convex object (a convex set or a convex function) a certain convex cone without loss of information about the object. From the duality theorem for cones duality theorems for other convex objects are deduced as consequences. The theme 'Duality formulae and the calculus of convex objects' is exhausted (from a certain precisely formulated point of view). Bibliography: 5 titles.

[en] The Dualized Standard Model which has a number of very interesting physical consequences is itself based on the concept of a nonabelian generalization to electric-magnetic duality. This paper explains first the reasons why the ordinary (Hodge) does not give duality for the nonabelian theory and then reviews the steps by which these difficulties are surmounted, leading to a generalized duality transform formulated in loop space. The significance of this in relation to the Dualized Standard Model is explained, and possibly also to some other areas. (author)

[en] The problem of wave-particle duality is considered within the framework of the algebraic approach. Contrary to the widespread belief, we demonstrate that wave-particle duality can be reconciled with the assumption that there exists some local physical reality determining the results of local measurements. A number of quantum experiments—double-slit electron scattering, Wheeler’s delayed choice experiment, the past of photons passed through the interferometer—are discussed using the concept of locality. A clear physical interpretation of these experiments that does not contradict classical concepts is provided.

[en] Most known papers on spectral synthesis in complex domains are based on transitions from problems of spectral synthesis to equivalent problems of local description. As a rule, such a transition is carried out in the framework of some special assumptions and meets considerable difficulties. A general method developed in this paper enables one to verify the duality theorem in the setting of several complex variables, when each operator π_p(D), p=1,...,q, acts with respect to a single variable. These assumptions cover the case of a system of partial differential operators. The duality transition here breaks into three separate steps. Two of them are connected with classical results of the theory of analytic functions and only one relates to general duality theory. This allows one to speak about singling out the analytic component of the transition from a spectral synthesis problem to an equivalent problem of local description