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In this note, we construct self-dual vortices and cosmic strings from the generalized Abelian Higgs theory in which the Higgs potential is a polynomial whose degree depends on the number m. When |m|>0 , we obtain sharp existence theorems for vortices and strings corresponding to the absence and presence of gravity, respectively, over the whole plane. In the absence of gravity, we prove the existence of vortices by using a monotone iteration method. When gravity is taken into account, a regularization method and a fixed-point theorem are used to show that multiple string solutions exist under a sufficient condition imposed only on the total number of strings. In addition, a series of properties with respect to vortices and strings have also been established. [ABSTRACT FROM AUTHOR]

In this paper, an intrinsic description of some families of linear codes with symmetries is given, showing that they can be described more generally as quasi-group codes, that is, as linear codes allowing a group of permutation automorphisms which acts freely on the set of coordinates. An algebraic description, including the concatenated structure, of such codes is presented. This allows to construct quasi-group codes from codes over rings, and vice versa. The last part of the paper is dedicated to the investigation of self-duality of quasi-group codes. [ABSTRACT FROM AUTHOR]

We examine a generalization of the usual self-duality equations for Yang–Mills theory when the color space admits a nontrivial involution. This involution allows us to construct a nontrivial twist which may be combined with the Hodge star to form a twisted self-dual curvature. We will construct a simple example of twisted self-duality for s u (2) ⊕ s u (2) gauge theory along with its explicit solutions, both in Euclidean and Minkowski backgrounds, and then dimensionally reduce from four dimensions to obtain families of nontrivial nonlinear equations in lower dimensions. This twisted self-duality constraint will be shown to arise in E 7 exceptional field theory through a Scherk–Schwarz reduction and we will show how an Eguchi–Hanson gravitational instanton also obeys the twisted self-duality condition. [ABSTRACT FROM AUTHOR]

The existence of interacting higher-spin theories is tightly constrained by many no-go theorems. In this paper, we construct a chiral, higher-spin generalization of Yang–Mills theory in flat space which avoids these no-go theorems and has non-trivial tree-level scattering amplitudes with some higher-spin external legs. The fields and action are complex, so the theory is non-unitary and parity-violating, yet we find surprisingly compact formulae for all-multiplicity tree-level scattering amplitudes in the maximal helicity violating (MHV) sector, where the two negative helicity particles have identical but arbitrary spin. This is possible because the theory admits a perturbative expansion around its self-dual sector. Using twistor theory, we prove the classical integrability of this self-dual sector and show that it can be described on spacetime by an infinite tower of interacting massless scalar fields. We also give a twistor construction of the full theory and use it to derive the formula for the MHV amplitude. [ABSTRACT FROM AUTHOR]

The construction of amplitudes on curved space-times is a major challenge, particularly when the background has non-constant curvature. We give formulae for all tree-level graviton scattering amplitudes in curved self-dual (SD) radiative space-times; these are chiral, source-free, asymptotically flat spaces determined by free characteristic data at null infinity. Such space-times admit an elegant description in terms of twistor theory, which provides the powerful tools required to exploit their underlying integrability. The tree-level S-matrix is written in terms of an integral over the moduli space of holomorphic maps from the Riemann sphere to twistor space, with the degree of the map corresponding to the helicity configuration of the external gravitons. For the MHV sector, we derive the amplitude directly from the Einstein–Hilbert action of general relativity, while other helicity configurations arise from a natural family of generating functionals and pass several consistency checks. The amplitudes in SD radiative space-times exhibit many novel features that are absent in Minkowski space, including tail effects. There remain residual integrals due to the functional degrees of freedom in the background space-time, but our formulae have many fewer such integrals than would be expected from space-time perturbation theory. In highly symmetric special cases, such as SD plane waves, the number of residual integrals can be further reduced, resulting in much simpler expressions for the scattering amplitudes. [ABSTRACT FROM AUTHOR]

In this paper, we provide a necessary and sufficient condition for a proper subdual cone to be self-dual. As an application of this, we illustrate self-duality of proper cones that occur naturally in the theory of optimization. [ABSTRACT FROM AUTHOR]

In this paper, we construct a self-dual complete resolution of a module defined by a pair of embedded complete intersection ideals in a local ring. Our construction is based on a gluing construction of Herzog and Martsinkovsky and exploits the structure of Koszul homology in the embedded complete intersection case. As a consequence of our construction, we produce an isomorphism between certain stable homology and cohomology modules. [ABSTRACT FROM AUTHOR]

An antinorm is a concave nonnegative homogeneous functional on a convex cone. It is shown that if the cone is polyhedral, then every antinorm has a unique continuous extension from the interior of the cone. The main facts of the duality theory in convex analysis, in particular, the Fenchel-Moreau theorem, are generalized to antinorms. However, it is shown that the duality relation for antinorms is discontinuous. In every dimension, there are infinitely many self-dual antinorms on the positive orthant and, in particular, infinitely many autopolar polyhedra. For the two-dimensional case, we characterize them all. The classification in higher dimensions is left as an open problem. Applications to linear dynamical systems, to the Lyapunov exponent of random matrix products, to the lower spectral radius of nonnegative matrices, and to convex trigonometry are considered. [ABSTRACT FROM AUTHOR]

Motivated by a recent work of Chen–Zheng (J Geom Anal 32:141, 2022) on Strominger space forms, we prove that a compact Hermitian surface with pointwise constant holomorphic sectional curvature with respect to a Gauduchon connection ∇ t is either Kähler, or an isosceles Hopf surface with an admissible metric and t = - 1 or t = 3 . In particular, a compact Hermitian surface with pointwise constant Lichnerowicz holomorphic sectional curvature is Kähler. We further generalize the result to the case for the two-parameter canonical connections introduced by Zhao–Zheng (On Gauduchon Kähler-like manifolds. ArXiv: 2108.08181), which extends a previous result by Apostolov–Davidov–Muškarov (Trans Am Math Soc 348:3051–3063, 1996). [ABSTRACT FROM AUTHOR]

В статье рассматриваются возможности применения библиотеки Logic системы компьютерной алгебры Maple в аспекте компьютерного моделирования логических схем в различных базисах. Смоделированы основные логические элементы в Maple. На конкретном примере детально представлен алгоритм построения логической схемы в различных базисах.

Resistivity saturation is found on both superconducting and insulating sides of an "avoided" magnetic-field-tuned superconductor-to-insulator transition (H-SIT) in a two-dimensional In/InOx composite, where the anomalous metallic behavior cuts off conductivity or resistivity divergence in the zero-temperature limit. The granular morphology of the material implies a system of Josephson junctions (JJs) with a broad distribution of Josephson coupling EJ and charging energy EC, with an H-SIT determined by the competition between EJ and EC. By virtue of self-duality across the true H-SIT, we invoke macroscopic quantum tunneling effects to explain the temperatureindependent resistance where the "failed superconductor" side is a consequence of phase fluctuations and the "failed insulator" side results from charge fluctuations. While true self-duality is lost in the avoided transition, its vestiges are argued to persist, owing to the incipient duality of the percolative nature of the dissipative path in the underlying random JJ system. [ABSTRACT FROM AUTHOR]

We investigate the two-variable Wasserstein mean of positive definite operators, as a unique positive solution of the nonlinear equation obtained from the gradient of the objective function of the least squares problem. A various properties of two-variable Wasserstein mean including the symmetry and the refinement of the self-duality are shown. Furthermore, interesting inequalities such as the Ando–Hiai inequality and bounds for the difference between the two-variable arithmetic and Wasserstein mean are provided. Finally, we explore the relationship between the tolerance relation and two-variable Wasserstein mean of positive definite Hermitian matrices. [ABSTRACT FROM AUTHOR]

One of the main aims of the paper is to develop the mesh geometry technique for corank-two edge-bipartite graphs Δ with n + 2 ≥ 3 vertices, and the mesh algorithms introduced in [30,33] and successfully studied in our recent article [42]. We introduce and study the concept of a self-duality of mesh geometries Γ ( R ˆ Δ , Φ Δ) viewed as Φ Δ -mesh translation quivers. We show how self-dualities of mesh geometries Γ ( R ˆ Δ , Φ Δ) and the mesh geometry technique is applied to an affirmative algorithmic solution of so called Horn-Sergeichuk type problem [9, Problem 4.3] on the self-congruency of square integer matrices A ∈ M n + 2 (Z) , for the class of non-symmetric Gram matrices A = G ˇ Δ of corank-two loop-free edge-bipartite graphs Δ, with n + 2 ≤ 6 vertices. More precisely, we show that each of the mesh geometries Γ ( R ˆ Δ , Φ Δ) is self-dual, we construct its dual form Γ ( R ˆ Δ , Φ Δ) o p = Γ ( R ˆ Δ , Φ Δ − 1) isomorphic with Γ ( R ˆ Δ , Φ Δ) , and we construct a canonical self-duality isomorphism f Δ : Γ ( R ˆ Δ , Φ Δ) → Γ ( R ˆ Δ , Φ Δ) o p of mesh translation quivers. Using the self-duality f Δ we construct combinatorial algorithms such that, given a square Gram matrix A = G ˇ Δ ∈ M n + 2 (Z) of Δ lying in this class, they are able to compute a Z -invertible matrix B ∈ M n + 2 (Z) that coincide with its inverse B − 1 and defines the congruence of A with A t r , i.e., the equation A t r = B t r ⋅ A ⋅ B is satisfied. An idea of our solution is outlined in Section 8 of our recent article [42] , where among others two of our 13 algorithms solving the problem are constructed. The remaining 11 algorithms are constructed in the present article. We do it by means of the structure of the standard self-dual Φ Δ -mesh translation quiver Γ ( R ˆ Δ , Φ Δ) (called a geometry) canonically associated with Δ, consisting of Φ Δ -meshes of Φ Δ -orbits O Δ (w) of vectors w ∈ R ˆ Δ ⊆ Z n + 2 , where Φ Δ : Z n + 2 → Z n + 2 is the Coxeter transformation of Δ. We construct in the paper such self-dual Φ Δ -mesh geometry Γ ( R ˆ Δ , Φ Δ) , for each of the corank-two loop-free edge-bipartite graphs Δ, with n + 2 ≤ 6 vertices. [ABSTRACT FROM AUTHOR]

In this paper, we study the (anti-)self-duality equations ∗ F ∧ F = ± F ∧ F in the eight-dimensional Euclidean space. Using properties of the Clifford algebra C l 0 , 8 (R) , we find a new solution to these equations. [ABSTRACT FROM AUTHOR]

On a 4-manifold of conformal torsion-free connection with zero signature (--++) we found conditions under which the conformal curvature matrix is dual (self-dual or anti-self-dual). These conditions are 5 partial differential equations of the 2nd order on 10 coefficients of the angular metric and 4 partial differential equations of the 1st order, containing also 3 coefficients of external 2-form of charge. (External 2-form of charge is one of the components of the conformal curvature matrix.) Duality equations for a metric of a diagonal type are composed. They form a system of five second-order differential equations on three unknown functions of all four variables. We found several series of solutions for this system. In particular, we obtained all solutions for a logarithmically polynomial diagonal metric, that is, for a metric whose coefficients are exponents of polynomials of four variables.

We present three versions of the Lax–Milgram theorem in the framework of Hilbert C ∗ -modules, two for self-dual ones over W ∗ -algebras and one for those over C ∗ -algebras of compact operators. It is remarkable that while the Riesz theorem is not valid for certain Hilbert C ∗ -modules over C ∗ -algebras of compact operators, however, the modular Lax–Milgram theorem turns out to be valid for all of them. We also give several examples to illustrate our results, in particular, we show that the main theorem is not true for Hilbert modules over arbitrary C ∗ -algebras. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we determine the automorphism group of the p -cones (p ≠ 2) in dimension greater than two. In particular, we show that the automorphism group of those p -cones are the positive scalar multiples of the generalized permutation matrices that fix the main axis of the cone. Next, we take a look at a problem related to the duality theory of the p -cones. Under the Euclidean inner product it is well-known that a p -cone is self-dual only when p = 2. However, it was not known whether it is possible to construct an inner product depending on p which makes the p -cone self-dual. Our results show that no matter which inner product is considered, a p -cone will never become self-dual unless p = 2 or the dimension is less than three. [ABSTRACT FROM AUTHOR]

Abstract For certain purposes it would be of interest to find aggregation functions for which the aggregate value of the shortfalls coincides with the shortfall of the aggregate value of the original attainments, i.e. self-dual aggregation functions. It is crucial to know the scale type in which attainments are measured and how to reverse that scale to measure the corresponding shortfalls properly. In this paper we consider alternative definitions of shortfalls depending on the scale type and extend some characterisation results to this new framework. [ABSTRACT FROM AUTHOR]

In this paper, Seiberg Witten like equations are constructed on 7 manifolds endowed with G2-structure, lifted by SU(3)-structure. Then a global solution is obtained on the strictly-Pseudoconvex CR 7-manifolds for a given negative and constant scalar curvature. [ABSTRACT FROM AUTHOR]

Partially massless theory in three dimensions is revisited and its relationship with the self-dual massive gravity is considered. The only mode of the partially massless theory is shown explicitly through an action for a scalar field on (A)dS background. This fact can be generalized to higher dimensions. This degree of freedom is altered when a triadic Chern-Simons is introduced, giving rise to the self-dual massive gravity on (A)dS background. We present another physical system with partially massless symmetry and its connection with topologically massive gravity is discussed. [ABSTRACT FROM AUTHOR]

In electromagnetic systems, duality is manifested in various forms: circuit, Keller–Dykhne, electromagnetic, and Babinet dualities. These dualities have been developed individually in different research fields and frequency regimes, leading to a lack of unified perspective. In this paper, we establish a unified view of these dualities in electromagnetic systems. The underlying geometrical structures behind the dualities are elucidated by using concepts from algebraic topology and differential geometry. Moreover, we show that seemingly disparate phenomena, such as frequency-independent effective response, zero backscattering, and critical response, can be considered to be emergent phenomena of self-duality.

We review aspects of twistor theory, its aims and achievements spanning the last five decades. In the twistor approach, space--time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold--the twistor space. After giving an elementary construction of this space, we demonstrate how solutions to linear and nonlinear equations of mathematical physics--anti-self-duality equations on Yang--Mills or conformal curvature--can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang--Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang--Mills equations, and Einstein--Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally, we discuss the Newtonian limit of twistor theory and its possible role in Penrose's proposal for a role of gravity in quantum collapse of a wave function. [ABSTRACT FROM AUTHOR]

We reconsider the discrete dual of the immediate exchange model and define a more general class of models where mass is split, exchanged and merged. We relate the splitting process to the symmetric inclusion process via thermalization and from that obtain symmetries and self-duality for it and its generalization. We show that analogous properties hold for models where the splitting is related to the symmetric exclusion process or to independent random walkers. [ABSTRACT FROM AUTHOR]

For a d-dimensional convex polytope $$P\subset {\mathbb {R}}^d$$ , the outer normal unit vectors of its facets span another d-dimensional polytope, which is called the outer normal transform of P and denoted by $$P^*$$ . It seems that this transform, which clearly differs from the usual polarity transform, was not seriously investigated until now. We derive results about polytopes having natural properties with respect to this transform, like self-duality or equality for the composition of it. Among other things we prove that if P is a convex polytope inscribed in the unit sphere with the property that the circumradii of its facets are all equal, then $$(P^*)^*$$ coincides with P. We also prove that the converse is true when P or $$P^*$$ has at most 2 d vertices. For the three-dimensional case, we characterize the family of those tetrahedra T such that T and $$T^*$$ are congruent. [ABSTRACT FROM AUTHOR]

In this work, over n-dimensional (n > 4) oriented inner product space V, duality operator TΦ is defined by using the non zero ,(n-4)-form Φ and it is shown that this operator is symmetric. By the means of operator TΦ , Over the spaces whose dimension is greater than four we defined self-duality, anti-self-duality, weak self-duality and weak anti-self-duality of a 2-form. Especially, over the spaces Rn for 5<=n<=8 the duality operator TΦ which corresponds to the fundamental forms is studied in details. [ABSTRACT FROM AUTHOR]

Seiberg--Witten equations, which are defined on any 4 manifolds, consist of two equations [3, 6, 7]. The first of these equations is called Dirac equation and the latter curvature equation. In higher dimensions, generalized self--duality is used to describe Seiberg--Witten equations [1, 6, 4]. In this paper, Seiberg--Witten equations were obtained by choosing different self duality and by means of Spinc--structure which was given in [5]. Then, non trivial solutions are given by choosing different self--duality. [ABSTRACT FROM AUTHOR]

We compute the quotient of the self-duality equation for conformal metrics by the action of the diffeomorphism group. We also determine Hilbert polynomial, counting the number of independent scalar differential invariants depending on the jet-order, and the corresponding Poincaré function. We describe the field of rational differential invariants separating generic orbits of the diffeomorphism pseudogroup action, resolving the local recognition problem for self-dual conformal structures. [ABSTRACT FROM AUTHOR]

We characterize the module categories of suitably finite Hopf algebroids (more precisely, ×R-bialgebras in the sense of Takeuchi (1977) that are Hopf and finite in the sense of a work by the author (2000)) as those k-linear abelian monoidal categories that are module categories of some algebra, and admit dual objects for "sufficiently many" of their objects. Then we proceed to show that in many situations the Hopf algebroid can be chosen to be self-dual, in a sense to be made precise. This generalizes a result of Pfeiffer for pivotal fusion categories and the weak Hopf algebras associated to them. [ABSTRACT FROM AUTHOR]