Non-trivial Identities Research Articles

Computing degree based topological indices of algebraic hypergraphs

Topological indices are numerical parameters that indicate the topology of graphs or hypergraphs. A hypergraph H=(V(H),E(H)) consists of a vertex set V(H) and an edge set E(H), where each edge e∈E(H) is a subset of V(H) with at least two elements. In this paper, our main aim is to introduce a general hypergraph structure for the prime ideal sum (PIS)- graph of a commutative ring. The prime ideal sum hypergraph of a ring R is a hypergraph whose vertices are all non-trivial ideals of R and a subset of vertices Ei with at least two elements is a hyperedge whenever I+J is a prime ideal of R for each non-trivial ideal I, J in Ei and Ei is maximal with respect to this property. Moreover, we also compute some degree-based topological indices (first and second Zagreb indices, forgotten topological index, harmonic index, Randić index, Sombor index) for these hypergraphs. In particular, we describe some degree-based topological indices for the newly defined algebraic hypergraph based on prime ideal sum for Zn where n=pα,pq,p2q,p2q2,pqr,p3q, p2qr,pqrs for the distinct primes p,q,r and s.

Functional renormalization group for “p = 2” like glassy matrices in the planar approximation II. Ward identities method in the deep IR

This paper, as a continuation of our previous investigation [Nucl. Phys. B 1005 (2024) 116582] aims to study the glassy random matrices with quenched Wigner disorder. In this previous work, we have constructed a renormalization group based on the effective deterministic kinetic spectrum emerging from large N limit, and we extended approximate solutions using standard vertex expansion, at the leading order of the derivative expansion. Now in the following work, by introducing the non-trivial Ward identities which come from the (U(N))×2 symmetry broken of the effective kinetic action, we provide in the deep IR the explicit solution of the functional renormalization group for a model with quartic coupling by solving the Hierarchy to all orders in the local sector, which in particular imply the vanishing of the anomalous dimension. The numerical investigations confirm the first-order phase transition discovered in the vertex expansion framework, both in the active and passive schemes. Finally, we extend the discussion to hermitian matrices.

Order Ideals on Lexicographic Direct Sum of Three Totally Ordered Abelian Groups

Order ideals play an important role in the study of abstract algebra, especially in the study of ordered groups. In this paper, we focus on the study of order ideals in lexicographic direct sums of totally ordered Abelian groups. We begin by examining the order ideals in the group of integers , and the group of real numbers It is shown that there are no non-trivial order ideals in both groups. Next, we revisit the order ideals in the lexicographic direct sum of two totally ordered Abelian groups, The only non-trivial order ideal of is Furthermore, our study extends to the lexicographic direct sum of three totally ordered Abelian groups: and We investigate the non-trivial order ideals in these structures. It is stated that the non-trivial order ideals of are only and Furthermore, the non-trivial order ideals of are only and .Keywords: order ideal; lexicographic order; direct sum.

S-duality in the Cardy-like limit of the superconformal index

We evaluate the superconformal index of 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM with gauge algebra so(2Nc + 1) in the Cardy-like limit. We then study the relation with the results obtained for the S-dual usp(2Nc), discussing the fate of S-duality in different regions of charges. We find that S-duality is preserved thanks to a non-trivial integral identity that relates the three sphere partition functions of pure 3d Chern-Simons gauge theories.

Graded identities with involution for the algebra of upper triangular matrices

Let F be a field of characteristic zero and let m≥2 be an integer. In this paper, we prove that if a group grading on UTm(F) admits a graded involution then this grading is a coarsening of a Z⌊m2⌋-grading on UTm(F) and the graded involution is equivalent to the reflection or symplectic involution on UTm(F), this grading is called the finest grading on UTm(F). Furthermore, if m≤4 the algebra UTm(F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z⌊m2⌋,⁎)-identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z⌊m2⌋,⁎)-codimensions is determined. As a consequence, we prove that for any G-grading on UTm(F) and any graded involution the (G,⁎)-exponent is m. Finally, we study the algebra UT3(F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z-grading and the Z2-grading induced by (0,1,0). We determine a basis for the (Z2,⁎)-identities and we compute the codimension sequence for the (Z2,⁎)-graded identities for UT3(F).

Double copy for tree-level form factors. Part II. Generalizations and special topics

Both the Bern, Carrasco, and Johansson (BCJ) and the Kawai, Lewellen, and Tye (KLT) double-copy formalisms have been recently generalized to a class of scattering matrix elements (so-called form factors) that involve local gauge-invariant operators. In this paper, we continue the study of double copy for form factors. First, we generalize the double-copy prescription to form factors of higher-length operators tr(ϕm) with m ≥ 3. These higher-length operators introduce new non-trivial color identities, but the double-copy prescription works perfectly well. The closed formulae for the CK-dual numerators are also provided. Next, we discuss the υ→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overrightarrow{\\upsilon} $$\\end{document} vectors which are central ingredients appearing in the factorization relations of both the KLT kernels and the gauge form factors. We present a general construction rule for the υ→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overrightarrow{\\upsilon} $$\\end{document} vectors and discuss their universal properties. Finally, we consider the double copy for the form factor of the tr(F2) operator in pure Yang-Mills theory. In this case, we propose a new prescription which involves a gauge invariant decomposition for the form factor and a mixture of different CK-dual numerators appearing in the expansion. The new prescription for the more complicated double copy has been verified up to five external gluons.

Logicism, Frege and the Logical Composition of Informative Identities

Informative, non-trivial identities (like the definition of number) form the core of Frege's logicist program. This is why it is important to be able to dissipate any impression that there might be something paradoxical about them. However, it is easy to have such an impression : informative identities seem to be able to have either correctness (because the two elements involved in the identity really are identical) or informativeness (because the two elements involved in the identity are not exactly identical). This problem is sometimes called ‘the paradox of analysis.’ The first step in the direction of a Fregean solution to the paradox is, I argue, the elucidation of the status of ‘fruitful definitions’ as informative identities in Frege.

Metric dimension in a prime ideal sum graph of a commutative ring

The prime ideal sum graph of a commutative unital ring [Formula: see text], denoted by [Formula: see text], is an undirect and simple graph whose vertices are nontrivial ideals of [Formula: see text] and there exists an edge between two distinct vertices if and only if their sum is a prime ideal of [Formula: see text]. In this paper, the metric dimension of [Formula: see text] is discussed and some formulae for this parameter in [Formula: see text] are given.

On the Inclusion Ideal Graph of Semigroups

The inclusion ideal graph [Formula: see text] of a semigroup [Formula: see text] is an undirected simple graph whose vertices are all the nontrivial left ideals of [Formula: see text] and two distinct left ideals [Formula: see text], [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. The purpose of this paper is to study algebraic properties of the semigroup [Formula: see text] as well as graph theoretic properties of [Formula: see text]. We investigate the connectedness of [Formula: see text] and show that the diameter of [Formula: see text] is at most 3 if it is connected. We also obtain a necessary and sufficient condition of [Formula: see text] such that the clique number of [Formula: see text] is the number of minimal left ideals of [Formula: see text]. Further, various graph invariants of [Formula: see text], viz. perfectness, planarity, girth, etc., are discussed. For a completely simple semigroup [Formula: see text], we investigate properties of [Formula: see text] including its independence number and matching number. Finally, we obtain the automorphism group of [Formula: see text].

In Search of Comrade Agronomof: Some Tribonacci History

A note from May 1914 by a certain Agronomof in the Belgian journal Mathesis is frequently cited in the context of the mathematical history of the Tribonacci number sequence. We round out the historical perspective by providing details on Agronomof’s life and work, and highlight two late-nineteenth-century occurrences of this number sequence. One occurrence is connected to Charles Darwin’s On the Origin of Species. Furthermore, we show that the main identity in Agronomof’s note can be leveraged to derive several nontrivial identities with surprising ease and elegance. Some of these identities have only recently been proven by different and more laborious methods; others are new.

Generic length functions on countable groups

Let L(G) denote the space of integer-valued length functions on a countable group G endowed with the topology of pointwise convergence. Assuming that G does not satisfy any non-trivial mixed identity, we prove that a generic (in the Baire category sense) length function on G is a word length and the associated Cayley graph is isomorphic to a certain universal graph U independent of G. On the other hand, we show that every comeager subset of L(G) contains 2ℵ0 “asymptotically incomparable” length functions. A combination of these results yields 2ℵ0 pairwise non-equivalent regular representations G→Aut(U). We also prove that generic length functions are virtually indistinguishable from the model-theoretic point of view. Topological transitivity of the action of G on L(G) by conjugation plays a crucial role in the proof of the latter result.

On the intersection ideal graph of semigroups

The intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if their intersection is nontrivial. In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected, then the diameter of Γ(S) is at most two. Further, we classify the semigroups S in terms of their ideals such that the diameter of Γ(S) is two. We obtain the domination number, independence number, girth and the strong metric dimension of Γ(S). We have also investigated the completeness, planarity and perfectness of Γ(S). We show that if S is a completely simple semigroup, then Γ(S) is weakly perfect. More over, in this article, we give an upper bound of the chromatic number of Γ(S). Finally, if S is the union of n minimal left ideals, then we obtain the metric dimension and the automorphism group of Γ(S).

Elliptic modular graphs, eigenvalue equations and algebraic identities

We obtain eigenvalue equations satisfied by various elliptic modular graphs with five links where two of the vertices are unintegrated. Solving them leads to several nontrivial algebraic identities between these graphs.

Class of Crosscap Two Graphs Arising from Lattices–I

Let L be a lattice. The annihilating-ideal graph of L is a simple graph whose vertex set is the set of all nontrivial ideals of L and whose two distinct vertices I and J are adjacent if and only if I∧J=0. In this paper, crosscap two annihilating-ideal graphs of lattices with at most four atoms are characterized. These characterizations provide the classes of multipartite graphs, which are embedded in the Klein bottle.

The lambda extensions of the Ising correlation functions

We revisit, with a pedagogical heuristic motivation, the lambda extension of the low-temperature row correlation functions of the two-dimensional Ising model. In particular, using these one-parameter series to understand the deformation theory around selected values of , namely with m and n integers, we show that these series yield perturbation coefficients, generalizing form factors, that are D-finite functions. As a by-product these exact results provide an infinite number of highly non-trivial identities on the complete elliptic integrals of the first and second kind. These results underline the fundamental role of Jacobi theta functions and Jacobi forms, the previous D-finite functions being (relatively simple) rational functions of Jacobi theta functions, when rewritten in terms of the nome of elliptic functions.

Polynomial identities in Novikov algebras

In this paper, we study Novikov algebras satisfying nontrivial identities. We show that a Novikov algebra over a field of zero characteristic that satisfies a nontrivial identity satisfies some unexpected “universal” identities, in particular, right associator nilpotence, and right nilpotence of the commutator ideal. This, in particular, implies that a Novikov algebra over a field of zero characteristic satisfies a nontrivial identity if and only if it is Lie-solvable. We also establish that any system of identities of Novikov algebras over a field of zero characteristic follows from finitely many of them, and that the same holds over any field for multilinear Novikov identities. Some analogous simpler statements are also proved for commutative differential algebras.

Reduced (Twisted) Group C*-Algebras without Nontrivial Ideals

Reduced (Twisted) Group C*-Algebras without Nontrivial Ideals

Symbology for elliptic multiple polylogarithms and the symbol prime

Elliptic multiple polylogarithms occur in Feynman integrals and in particular in scattering amplitudes. They can be characterized by their symbol, a tensor product in the so-called symbol letters. In contrast to the non-elliptic case, the elliptic letters themselves satisfy highly non-trivial identities, which we discuss in this paper. Moreover, we introduce the symbol prime, an analog of the symbol for elliptic symbol letters, which makes these identities manifest. We demonstrate its use in two explicit examples at two-loop order: the unequal-mass sunrise integral in two dimensions and the ten-point double-box integral in four dimensions. Finally, we also report the result of the polylogarithmic nine-point double-box integral, which arises as the soft limit of the ten-point integral.

Majorana propagator on de Sitter space

We study the dynamics of Majorana fermions in an expanding de Sitter space and to that aim, by making use of the method of mode sums, we construct the vacuum Feynman propagator for Majorana fermions in de Sitter space. The Majorana condition implies nontrivial identities for the mode functions, which we carefully implement. We note that, under charge conjugation, the propagator transforms to minus itself, but do not discuss in detail the topological implications of this observation. We construct the propagator for a general complex mass and find that it is identical to the corresponding Dirac propagator, meaning that the coupling of fermions to the classical de Sitter gravitational background does not violate charge symmetry, i.e. it respects the Majorana condition. The complex mass propagator differs from its real mass counterpart in that the usual positive and negative energy projectors acquire a chiral rotation which depends on the phase of the mass term. We use our propagator to calculate the one-loop effective action for Majorana fermions, and find that it differs from that of Dirac fermions by a factor of 1/2, which accounts for the reduced number of degrees of freedom of the Majorana particle. Finally, we derive the Majorana and Dirac propagators for mixing n fermionic flavors and show that the number of CP-violating phases for Dirac fermions exceeds that of the Majorana fermions by {nleft( n-1right) }/{2}. This is vital for understanding how the dynamics of Dirac and Majorana fermions generate different CP-violating effects in the multi-flavor case.

On the genus of reduced cozero-divisor graph of commutative rings

Let R be a commutative ring with identity and let (x) be the principal ideal generated by \(x\in R.\) Let \(\varOmega (R)^*\) be the set of all nontrivial principal ideals of R. The reduced cozero-divisor graph \(\varGamma _r(R)\) of R is an undirected simple graph with \(\varOmega (R)^*\) as the vertex set and two distinct vertices (x) and (y) in \(\varOmega (R)^*\) are adjacent if and only if \((x)\nsubseteq (y)\) and \((y)\nsubseteq (x)\). In this paper, we characterize all classes of commutative Artinian non-local rings for which the reduced cozero-divisor graph has genus at most one.