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Motivated by the work of A. Zelevinsky on positive self-adjoint Hopf (PSH) algebras, we define what we call a symmetric self-adjoint Hopf (SSH) structure for a certain kind of semisimple abelian categories. It is known that every PSH algebra admits a natural action of the associated Heisenberg double. We construct canonical morphisms lifting the relations that define this action on the algebra level, and define an object that we call a categorical Heisenberg double that is a natural setting for considering these morphisms. As examples, we exhibit the SSH structure on the categories of polynomial functors and equivariant polynomial functors. In the case of the category of polynomial functors, we obtain categorification of the Fock space representation of the infinite-dimensional Heisenberg algebra. [ABSTRACT FROM AUTHOR]

In recent years, string solvers have become an essential component in many formal verification, security analysis, and bug-finding tools. Such solvers typically support a theory of string equations, the length function, and the regular-expression membership predicate. These enable considerable expressive power, which comes at the cost of slow solving time, and in some cases even non-termination. We present three techniques, designed for word-based SMT string solvers, to mitigate these problems: (1) detecting overlapping variables, which is essential to avoiding common cases of non-termination; (2) pruning of the search space via bi-directional integration between the string and integer theories, enabling new cross-domain heuristics; and (3) a binary search based heuristic, allowing the procedure to skip unnecessary string length queries and converge on consistent length assignments faster for large strings. We have implemented above techniques atop the Z3-str solver, resulting in a significantly more robust and efficient solver, dubbed Z3str2, for the quantifier-free theory of string equations, the regular-expression membership predicate, and linear arithmetic over the length function. We report on a series of experiments over four sets of challenging real-world benchmarks, where we compare Z3str2 with five different string solvers: S3, CVC4, Kaluza, PISA and Stranger. Each of these tools utilizes a different solving strategy and/or string representation (based e.g. on words, bit vectors or automata). The results point to the efficacy of our proposed techniques, which yield dramatic performance improvement. We also demonstrate performance improvements enabled by Z3str2 in the context of symbolic execution for string-manipulating programs. We observe that the techniques presented here are of broad applicability, and can be integrated into other string solvers to improve their performance. [ABSTRACT FROM AUTHOR]

We study parallel generalized retracts used for the construction of isomorphisms between free (Abelian) topological groups and free locally convex spaces. [ABSTRACT FROM AUTHOR]

The superconformal index on ℝℙ × S can be derived exactly by the localization technique and applied to the direct proof of Abelianmirror symmetry. We find two sets of parity conditions compatible with the unorientable property of ℝℙ and then rigorously show two kinds of Abelian mirror symmetry via the index on ℝℙ × S. [ABSTRACT FROM AUTHOR]

This paper is concerned with the upper bound of the number of limit cycles in unfolding of codimension 3 planar singularities with nilpotent linear parts. After making a central rescaling, the problem reduces to a perturbation problem of a one-parameter family of quadratic reversible systems. As the parameter a ∈ ( − 1 , 1 ) ∖ { 0 } is rational, except the case a = − 2 3 , based on the Chebyshev criterion for Abelian integrals and a rationalizing transformation, the problem could be solved theoretically. To illustrate our approaches, two particular cases (corresponding to nilpotent codimension 3 saddle and elliptic case respectively) are proved where the upper bound of the number of limit cycles is two. [ABSTRACT FROM AUTHOR]

LetGbe a finite group andpa prime divisor of |G|. Denote by IBr(G) and LBr(G) the sets of irreduciblep-Brauer characters and linearp-Brauer characters ofG, respectively. It is known that IBr(G) = LBr(G) if and only ifP∈ Sylp(G) is normal andG/Pis abelian. We consider the local situation in this note. LetBbe ap-block ofG. We characterize the finite groupsGwith IBr(B) ⊆ LBr(G). And we get some results related to those of Holm, Willems and Isaacs, Smith. [ABSTRACT FROM PUBLISHER]

We define a non-abelian tensor product of multiplicative Lie rings. This is a new concept providing a common approach to the non-abelian tensor product of groups defined by Brown and Loday and to the non-abelian tensor product of Lie rings defined by Ellis. We also prove an analogue of Miller's theorem for multiplicative Lie rings. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the asymptotic behavior of one- dimensional modified Stockwell transform of a tempered distribution signal through the quasiasymptotic behavior at origin or infinity of the signal itself. More precisely, we give some Abelian results which mean that we derive the asymptotic properties of theS-transform of a tempered signal from the quasiasymptotic properties of the signal itself and we do also the opposite. So, we also give some Tauberian results which describe some quasiasymptotic properties of the tempered signal by means of the asymptotic properties of its Stockwell transform. [ABSTRACT FROM AUTHOR]

In this paper, we shall analyze a three dimensional supersymmetry theory with $$\mathcal {N} = 2$$ supersymmetry. We will analyze the quantization of this theory, in the presence of a boundary. The effective Lagrangian used in the path integral quantization of this theory, will be given by the sum of the gauge fixing term and the ghost term with the original classical Lagrangian. Even though the supersymmetry of this effective Lagrangian will also be broken due to the presence of a boundary, it will be demonstrated that half of the supersymmetry of this theory can be preserved by adding a boundary Lagrangian to the effective bulk Lagrangian. The supersymmetric transformation of this new boundary Lagrangian will exactly cancel the boundary term generated from the supersymmetric transformation of the effective bulk Lagrangian. We will analyze the Slavnov-Taylor identity for this $$\mathcal {N} = 2$$ Yang-Mills theory with a boundary. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells having orthogonal diagonals. We discuss discrete coverings, discrete exterior calculus, and discrete Abelian differentials. Our presentation includes several new notions and results such as branched coverings of discrete Riemann surfaces, the discrete Riemann–Hurwitz Formula, double poles of discrete one-forms and double values of discrete meromorphic functions that enter the discrete Riemann–Roch Theorem, and a discrete Abel–Jacobi map. [ABSTRACT FROM AUTHOR]

Similar to the linear Harbourne constant recently introduced in [2], we study the elliptic H-constants of P² and of Abelian surfaces. We also study the Harbourne indices of curves on these surfaces. In particular, we show that there are configurations of smooth plane cubic curves whose Harbourne indices are arbitrarily close to -4. Consequently, we obtain that the H-constant of any surface X is less than or equal to -4. Related to these problems, we moreover give a new inequality for the number and multiplicities of singularities of elliptic curves arrangements on Abelian surfaces, inequality which has a close similarity to the one of Hirzebruch for lines arrangements on the plane. [ABSTRACT FROM AUTHOR]

Let G be a polycyclic group and α a regular automorphism of order four of G. If the map φ: G → G defined by g = [ g, α] is surjective, then the second derived group of G is contained in the centre of G. Abandoning the condition on surjectivity, we prove that C ( α ) and G/[ G, α ] are both abelian-by-finite. [ABSTRACT FROM AUTHOR]

Suppose that A is an abelian category whose derived category D(A) has Hom sets and arbitrary (small) coproducts, let T be a (not necessarily classical) (n-)tilting object of A and let H be the heart of the associated t-structure on D(A). We show that there is a triangulated equivalence of unbounded derived categories D(H)≃-→ D(A) which is compatible with the inclusion functor H →D(A). The result admits a straightforward dualization to cotilting objects in abelian categories whose derived category has Hom sets and arbitrary products. [ABSTRACT FROM AUTHOR]

Let (X, x0) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for (X, x0) produces a homomorphism from the abelianization of the F-divided fundamental group scheme of X to the F-divided fundamental group of the Albanese variety of X. We prove that this homomorphism is surjective with finite kernel. The kernel is also described. [ABSTRACT FROM AUTHOR]

E. Rips constructed a series of groups such that their group rings have zero divisors. such groups can serve as counterexamples to Kaplansky's problem on zero divisors. The problem is to find such a group without torsion. We study simplest groups of this series, classify such groups, describe the structure, and show that all such groups have 2-torsion. [ABSTRACT FROM AUTHOR]

A gauge fixing condition is presented here for non-Abelian gauge theory on the manifold R ⊗ S 1 ⊗ S 1 ⊗ S 1 . It is proved that the new gauge fixing condition is continuous and free from the Gribov ambiguity. While perturbative calculations based on the new gauge condition behave like those based on the axial gauge in ultraviolet region, infrared behaviours of the perturbative series under the new gauge fixing condition are quite nontrivial. The new gauge condition, which reads n ⋅ ∂ n ⋅ A = 0 , may not satisfy the boundary condition A μ ( ∞ ) = 0 as required by conventional perturbative calculations for gauge theories on the manifold S 4 . However, such contradiction is not harmful for the theory considered here. [ABSTRACT FROM AUTHOR]

The infinitesimal symmetries of differential equations (DEs) or other geometric objects provide key insight into their analytical structure, including construction of solutions and of mappings between DEs. This article is a contribution to the algorithmic treatment of symmetries of DEs and their applications. Infinitesimal symmetries obey a determining system L of linear homogeneous partial differential equations, with the property that its solution vector fields form a Lie algebra L . We exhibit several algorithms that work directly with the determining system without solving it. A procedure is given that can decide if a system specifies a Lie algebra L , if L is abelian and if a system L ′ specifies an ideal in L . Algorithms are described that compute determining systems for transporter, Lie product and Killing orthogonal subspace. This gives a systematic calculus for Lie determining systems, enabling computation of the determining systems for normalisers, centralisers, centre, derived algebra, solvable radical and key series (derived series, lower/upper central series). Our methods thereby give algorithmic access to new geometrical invariants of the symmetry action. [ABSTRACT FROM AUTHOR]

In the present paper we evaluate the anomaly for the abelian axial current in a non abelian chiral gauge theory, by using dimensional regularization. This amount to formulate a procedure for managing traces with more than one γ . The suggested procedure obeys Lorentz covariance and cyclicity, at variance with previous approaches (e.g. the celebrated 't Hooft and Veltman's where Lorentz is violated). The result of the present paper is a further step forward in the program initiated by a previous work on the traces involving a single γ . The final goal is an unconstrained definition of γ in dimensional regularization. Here, in the evaluation of the anomaly, we profit of the axial current conservation equation, when radiative corrections are neglected. This kind of tool is not always exploited in field theories with γ , e.g. in the use of dimensional regularization of infrared and collinear divergences. [ABSTRACT FROM AUTHOR]

We show that dp-minimal valued fields are henselian and give classifications of dp-minimal ordered abelian groups and dp-minimal ordered fields without additional structure. [ABSTRACT FROM PUBLISHER]

We prove that the class of nilpotent by finite, solvable by finite, polycyclic by finite, nilpotent of nilpotency class n and supersolvable groups are closed under the formation of the non-abelian tensor product. We provide necessary and sufficient conditions for the non-abelian tensor product of finitely generated groups to be finitely generated. [ABSTRACT FROM AUTHOR]

A p-group G is called a C AC-p-group if CG(H)/H is cyclic for every non-cyclic abelian subgroup H in G with H ≰ Z(G). In this paper, we give a complete classification of finite C AC-p-groups. [ABSTRACT FROM AUTHOR]

We prove that abelian Yang–Baxter deformations of superstring coset σ models are equivalent to sequences of commuting TsT transformations, meaning T dualities and coordinate shifts. Our results extend also to fermionic deformations and fermionic T duality, and naturally lead to a TsT subgroup of the superduality group OSp ( d b , d b | 2 d f ) . In cases like AdS 5 × S 5 , fermionic deformations necessarily lead to complex models. As an illustration of inequivalent deformations, we give all six abelian deformations of AdS 3 . We comment on the possible dual field theory interpretation of these (super-) TsT models. [ABSTRACT FROM AUTHOR]

Lovelock theory of gravity provides a tractable model to investigate the effects of higher-curvature terms in the context of AdS/CFT. Yielding second order, ghost-free field equations, this theory represents a minimal setup in which higher-order gravitational couplings in asymptotically Anti-de Sitter (AdS) spaces, including black holes, can be solved analytically. This however has an obvious limitation as in dimensions lower than seven, the contribution from cubic or higher curvature terms is merely topological. Therefore, in order to go beyond quadratic order and study higher terms in AdS5 analytically, one is compelled to look for other toy models. One such model is the so-called quasi-topological gravity, which, despite being a higher-derivative theory, provides a tractable setup with R³ and R4 terms. In this paper, we investigate AdS5 black holes in quasi-topological gravity. We consider the theory conformally coupled to matter and in presence of Abelian gauge fields. We show that charged black holes in AdS5 which, in addition, exhibit a backreaction of the matter fields on the geometry can be found explicitly in this theory. These solutions generalize the black hole solution of quasi-topological gravity and exist in a region of the parameter spaces consistent with the constraints coming from causality and other consistency conditions. They have finite conserved charges and exhibit non-trivial thermodynamical properties. [ABSTRACT FROM AUTHOR]

In the first part of our paper, we show that there exist non-isomorphic derived equivalent genus 1 curves, and correspondingly there exist non-isomorphic moduli spaces of stable vector bundles on genus 1 curves in general. Neither occurs over an algebraically closed field. We give necessary and sufficient conditions for two genus 1 curves to be derived equivalent, and we go on to study when two principal homogeneous spaces for an abelian variety have equivalent derived categories. We apply our results to study twisted derived equivalences of the form D b ( J , α ) ≃ D b ( J , β ) , when J is an elliptic fibration, giving a partial answer to a question of Căldăraru. [ABSTRACT FROM AUTHOR]

Two words u and v are k -abelian equivalent if for each word x of length at most k , x occurs equally many times as a factor in both u and v . The notion of k -abelian equivalence is an intermediate notion between the abelian equivalence and the equality of words. In this paper, we study the equivalence classes induced by the k -abelian equivalence, mainly focusing on the cardinalities of the classes. In particular, we are interested in the number of singleton k -abelian classes, i.e., classes containing only one element. We find a connection between the singleton classes and cycle decompositions of the de Bruijn graph. We show that the number of classes of words of length n containing one single element is of order O ( n N m ( k − 1 ) − 1 ) , where N m ( l ) = 1 l ∑ d | l φ ( d ) m l / d is the number of necklaces of length l over an m -ary alphabet. We conjecture that the upper bound is sharp. We also remark that, for k even and m = 2 , the lower bound Ω ( n N m ( k − 1 ) − 1 ) follows from an old conjecture on the existence of Gray codes for necklaces of odd length. We verify this conjecture for necklaces of length up to 15. [ABSTRACT FROM AUTHOR]

In the paper, it is established that every hereditary hyperradical formation is saturated, and a description of all hereditary hyperradical formations is given. [ABSTRACT FROM AUTHOR]

In this work we study mesonic excitations in a Quantum Field Theory dual to the non Abelian T-dual of AdS5 × S5, using a D6 brane probe on the Sfetsos-Thompson background. Before and after the duality, we observe interesting differences between the spectra and interpret them. The spectrum of masses and the interactions between mesonic excitations teach valuable lessons about the character of non-Abelian T-duality and its implications for Holography. The case of Abelian T-duality is also studied. [ABSTRACT FROM AUTHOR]

Using the irreducibility of a natural irreducible representation of the theta group of an ample line bundle on an abelian variety, we derive a bound for the number of n-torsion points that lie on a given theta divisor. We present also two alternate approaches to attacking the case n = 2. [ABSTRACT FROM AUTHOR]

We write x ≺ y when x and y are vectors with each element of x less than or equal to the corresponding element of y and P ( w ) for the Parikh vector of a word w . A word w has abelian period p if it has the form u 0 u 1 ⋯ u m u m + 1 with | u 0 | ≤ p , | u i | = p for i = 1 , … m and | u m + 1 | ≤ p , and P ( u 0 ) ≺ P , P ( u 0 ) = P for i = 1 , … , m and P ( u m + 1 ) ≺ P for some vector P . Blanchet-Sadri et al. conjectured that if a word has abelian periods pd and qd , where gcd ⁡ ( p , q ) = 1 , and length at least 2 p q d − 1 then the number of distinct letters appearing in the word is at most d , and that under certain conditions the bound may be reduced to 2 p q d − 2 . We prove their conjecture. [ABSTRACT FROM AUTHOR]

Given a word w and a Parikh vector P , an abelian run of period P in w is a maximal occurrence of a substring of w having abelian period P . Our main result is an online algorithm that, given a word w of length n over an alphabet of cardinality σ and a Parikh vector P , returns all the abelian runs of period P in w in time O ( n ) and space O ( σ + p ) , where p is the norm of P , i.e., the sum of its components. We also present an online algorithm that computes all the abelian runs with periods of norm p in w in time O ( n p ) , for any given norm p . Finally, we give an O ( n 2 ) -time offline randomized algorithm for computing all the abelian runs of w . Its deterministic counterpart runs in O ( n 2 log ⁡ σ ) time. [ABSTRACT FROM AUTHOR]

We consider, in the harmonic superspace approach, the six-dimensional N = ( 1 , 0 ) supersymmetric model of abelian gauge multiplet coupled to a hypermultiplet. The superficial degree of divergence is evaluated and the structure of possible one-loop divergences is analyzed. Using the superfield proper-time and background-field technique, we compute the divergent part of the one-loop effective action depending on both the gauge multiplet and the hypermultiplet. The corresponding counterterms contain the purely gauge multiplet contribution together with the mixed contributions of the gauge multiplet and hypermultiplet. We show that the theory is on-shell one-loop finite in the gauge multiplet sector in agreement with the results of [1] . The divergences in the mixed sector cannot be eliminated by any field redefinition, implying the theory to be UV divergent at one loop. [ABSTRACT FROM AUTHOR]

Biadjoint scalar theories are novel field theories that arise in the study of non-abelian gauge and gravity amplitudes. In this short paper, we present exact nonperturbative solutions of the field equations, and compare their properties with monopole-like solutions in non-abelian gauge theory. Our results may pave the way for nonperturbative studies of the double copy. [ABSTRACT FROM AUTHOR]

The Gribov problem in the presence of a background field is analyzed: in particular, we study the Gribov copies equation in the Landau–De Witt gauge as well as the semi-classical Gribov gap equation. As background field, we choose the simplest non-trivial one which corresponds to a constant gauge potential with non-vanishing component along the Euclidean time direction. This kind of constant non-Abelian background fields is very relevant in relation with (the computation of) the Polyakov loop but it also appears when one considers the non-Abelian Schwinger effect. We show that the Gribov copies equation is affected directly by the presence of the background field, constructing an explicit example. The analysis of the Gribov gap equation shows that the larger the background field, the smaller the Gribov mass parameter. These results strongly suggest that the relevance of the Gribov copies (from the path integral point of view) decreases as the size of the background field increases. [ABSTRACT FROM AUTHOR]

The first example of a torsion-free abelian group $(A,+,0)$ such that the quotient group of $A$ modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’, Publ. Math. Debrecen27 (1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from $2$ to $2^{\aleph _{0}}$ is presented. Ring multiplications on $p$-pure subgroups of the additive group of the ring of $p$-adic integers are investigated using only elementary methods. [ABSTRACT FROM PUBLISHER]

We study topological groups that can be defined as Polish, pro-countable abelian groups, as non-archimedean abelian groups or as quasi-countable abelian groups, i.e., Polish subdirect products of countable, discrete groups, endowed with the product topology. We characterize tame groups in this class, i.e., groups all of whose continuous actions on a Polish space induce a Borel orbit equivalence relation, and relatively tame groups, i.e., groups all of whose diagonal actions [ABSTRACT FROM AUTHOR]

We describe a connected component of the space of stability conditions on abelian threefolds, and on Calabi-Yau threefolds obtained as (the crepant resolution of) a finite quotient of an abelian threefold. Our proof includes the following essential steps: Important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall-crossing, closely related to the support property. [ABSTRACT FROM AUTHOR]

All connected Cayley graphs over Abelian groups are Hamiltonian. However, for Cayley graphs over non-Abelian groups, Chen and Quimpo prove in [2] that Cayley graphs over Hamiltonian groups (i.e, non- Abelian groups in which every subgroup is normal) are Hamiltonian. In this paper we discuss a few of the ideas which have been developed to establish the existence of Hamiltonian cycles and paths in the vertex induced subgraphs of Cayley graphs over non-Abelian groups. [ABSTRACT FROM AUTHOR]

We discuss some recent developments in the theory of abelian model categories. The emphasis is on the hereditary condition and applications to homotopy categories of chain complexes and stable module categories. [ABSTRACT FROM AUTHOR]

The problem of determining whether several finite automata accept a word in common is closely related to the well-studied membership problem in transformation monoids. We raise the issue of limiting the number of final states in the automata intersection problem. For automata with two final states, we show the problem to be $${\oplus}$$ L-complete or NP-complete according to whether a nontrivial monoid other than a direct product of cyclic groups of order 2 is allowed in the automata. We further consider idempotent commutative automata and (Abelian, mainly) group automata with one, two, or three final states over a singleton or larger alphabet, elucidating (under the usual hypotheses on complexity classes) the complexity of the intersection nonemptiness and related problems in each case. [ABSTRACT FROM AUTHOR]

I review the main results leading to Factorization of QCD amplitudes in momentum space and, in view of the analogue results in coordinate space, the one-loop jet function in coordinate space is computed and Landau's equations for the Abelian radiative corrections to it are studied. Furthermore, two of these radiative corrections are reduced in quadrature. [ABSTRACT FROM AUTHOR]

We prove that the radial subalgebra in free orthogonal quantum group factors is maximal abelian and mixing, and we compute the associated bimodule. The proof relies on new properties of the Jones–Wenzl projections and on an estimate of certain scalar products of coefficients of irreducible representations. [ABSTRACT FROM AUTHOR]

In this article we show that the use of Deligne–Beilinson cohomology in the context of the U ( 1 ) BF theory on a closed 3-manifold M yields a discrete Z N BF theory whose partition function is an abelian TV invariant of M . By comparing the expectation values of the U ( 1 ) and Z N holonomies in both BF theories we obtain a reciprocity formula. [ABSTRACT FROM AUTHOR]

We elaborate on how to build, in a systematic fashion, two-field Abelian extensions of the Born–Infeld Lagrangian. These models realize the non-trivial duality groups that are allowed in this case, namely U ( 2 ) , S U ( 2 ) and U ( 1 ) × U ( 1 ) . For each class, we also construct an explicit example. They all involve an overall square root and reduce to the Born–Infeld model if the two fields are identified, but differ in quartic and higher interactions. The U ( 1 ) × U ( 1 ) and S U ( 2 ) examples recover some recent results obtained with different techniques, and we show that the U ( 1 ) × U ( 1 ) model admits an N = 1 supersymmetric completion. The U ( 2 ) example includes some unusual terms that are not analytic at the origin of field space. [ABSTRACT FROM AUTHOR]

Our main goal is to introduce some integral-type generalizations of the cosine and sine equations for complex-valued functions defined on a group G that need not be abelian. These equations provide a joint generalization of many trigonometric type functional equations such as d'Alembert's, Cauchy's, Gajda's, Kannappan's and Van Vleck's equations. We prove that the continuous solutions for the first type and the central continuous solutions for the second one of these equations can be expressed in terms of characters, additive maps and matrix elements of irreducible, 2-dimensional representations of the group G. So the theory is part of harmonic analysis on groups. [ABSTRACT FROM AUTHOR]

Let $F$ be a field of characteristic $p\geq 0$ and $G$ any group. In this article, the Engel property of the group of units of the group algebra $FG$ is investigated. We show that if $G$ is locally finite, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G$ is locally nilpotent and $G^{\prime }$ is a $p$-group. Suppose that the set of nilpotent elements of $FG$ is finite. It is also shown that if $G$ is torsion, then ${\mathcal{U}}(FG)$ is an Engel group if and only if $G^{\prime }$ is a finite $p$-group and $FG$ is Lie Engel, if and only if ${\mathcal{U}}(FG)$ is locally nilpotent. If $G$ is nontorsion but $FG$ is semiprime, we show that the Engel property of ${\mathcal{U}}(FG)$ implies that the set of torsion elements of $G$ forms an abelian normal subgroup of $G$. [ABSTRACT FROM AUTHOR]

It is shown that, for any field $\mathbb{F}\subseteq \mathbb{R}$, any ordered vector space structure of $\mathbb{F}^{n}$ with Riesz interpolation is given by an inductive limit of a sequence with finite stages $(\mathbb{F}^{n},\mathbb{F}_{\geq 0}^{n})$ (where $n$ does not change). This relates to a conjecture of Effros and Shen, since disproven, which is given by the same statement, except with $\mathbb{F}$ replaced by the integers, $\mathbb{Z}$. Indeed, it shows that although Effros and Shen’s conjecture is false, it is true after tensoring with $\mathbb{Q}$. [ABSTRACT FROM AUTHOR]

We develop a method of proving nonnegativity of the coefficients of certain polynomials, also called reduced forms, defined by Kirillov in his quasi-classical Yang-Baxter algebra, its abelianization, and related algebras. It has been shown previously that the relations of the abelianization of the quasi-classical Yang-Baxter algebra, also called the subdivision algebra, encode ways of subdividing flow polytopes. In turn, these subdivisions can be represented as reduced forms, or as reduction trees. We use reduction trees in the subdivision algebra to construct canonical triangulations of flow polytopes which are shellable. We explain how a shelling of the canonical triangulation can be read off from the corresponding reduction tree in the subdivision algebra. We then introduce the notion of shellable reduction trees in the subdivision and related algebras and define h-polynomials of reduction trees. In the case of the subdivision algebra, the h-polynomials of the canonical triangulations of flow polytopes equal the h-polynomials of the corresponding reduction trees, which motivated our definition. We show that the reduced forms in various algebras, which can be read off from the leaves of the reduction trees, specialize to the shifted h-polynomials of the corresponding reduction trees. This yields a technique for proving nonnegativity properties of reduced forms. As a corollary we settle a conjecture of Kirillov. [ABSTRACT FROM AUTHOR]

Electric–magnetic dualities are equivalence between strong and weak coupling constants. A standard example is the exchange of electric and magnetic fields in an abelian gauge theory. We show three methods to perform electric–magnetic dualities in the case of the non-commutative U ( 1 ) gauge theory. The first method is to use covariant field strengths to be the electric and magnetic fields. We find an invariant form of an equation of motion after performing the electric–magnetic duality. The second method is to use the Seiberg–Witten map to rewrite the non-commutative U ( 1 ) gauge theory in terms of abelian field strength. The third method is to use the large Neveu Schwarz–Neveu Schwarz (NS–NS) background limit (non-commutativity parameter only has one degree of freedom) to consider the non-commutative U ( 1 ) gauge theory or D3-brane. In this limit, we introduce or dualize a new one-form gauge potential to get a D3-brane in a large Ramond–Ramond (R–R) background via field redefinition. We also use perturbation to study the equivalence between two D3-brane theories. Comparison of these methods in the non-commutative U ( 1 ) gauge theory gives different physical implications. The comparison reflects the differences between the non-abelian and non-commutative gauge theories in the electric–magnetic dualities. For a complete study, we also extend our studies to the simplest abelian and non-abelian p -form gauge theories, and a non-commutative theory with the non-abelian structure. [ABSTRACT FROM AUTHOR]

Richomme, Saari and Zamboni (2011) [39] proved that at every position of a Sturmian word starts an abelian power of exponent k for every k > 0 . We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words. We give a formula for computing the maximum exponent of an abelian power of abelian period m starting at a given position in any Sturmian word of rotation angle α . By considering all possible abelian periods m , we recover the result of Richomme, Saari and Zamboni. As an analogue of the critical exponent, we introduce the abelian critical exponent A ( s α ) of a Sturmian word s α of angle α as the quantity A ( s α ) = lim sup k m / m = lim sup k m ′ / m , where k m (resp. k m ′ ) denotes the maximum exponent of an abelian power (resp. of an abelian repetition) of abelian period m (the superior limits coincide for Sturmian words). We show that A ( s α ) equals the Lagrange constant of the number α . This yields a formula for computing A ( s α ) in terms of the partial quotients of the continued fraction expansion of α . Using this formula, we prove that A ( s α ) ≥ 5 and that the equality holds for the Fibonacci word. We further prove that A ( s α ) is finite if and only if α has bounded partial quotients, that is, if and only if s α is β -power-free for some real number β . Concerning the infinite Fibonacci word, we prove that: i ) The longest prefix that is an abelian repetition of period F j , j > 1 , has length F j ( F j + 1 + F j − 1 + 1 ) − 2 if j is even or F j ( F j + 1 + F j − 1 ) − 2 if j is odd, where F j is the j th Fibonacci number; ii ) The minimum abelian period of any factor is a Fibonacci number. Further, we derive a formula for the minimum abelian periods of the finite Fibonacci words: we prove that for j ≥ 3 the Fibonacci word f j , of length F j , has minimum abelian period equal to F ⌊ j / 2 ⌋ if j = 0 , 1 , 2 mod 4 or to F 1 + ⌊ j / 2 ⌋ if j = 3 mod 4 . [ABSTRACT FROM AUTHOR]