Unitary Structure of Palindromes in DNA

We investigate the quantum behavior encountered in palindromes within DNA structure. In particular we reveal the unitary structure of usual palindromic sequences found in genomic DNAs of all living organisms using the Schwinger approach. We clearly demonstrate the role played by palindromic configurations with special emphasis on physical symmetries in particular subsymmetries of unitary structure. We unveil the prominence of unitary structure in palindromic sequences in the sense that vitally significant information endowed within DNA could be transformed unchangeably in the process of transcription. We introduce a new symmetry relation namely purine-purine or pyrimidine-pyrimidine symmetries (p-symmetry) in addition to the already known symmetry relation of purine-pyrimidine symmetries (pp symmetry) given by Chargaff rule. Therefore important vital functions of a living organisms are protected by means of these symmetric features. It is understood that higher order palindromic sequences could be generated in terms of the basis of the highest prime numbers that make up the palindrome sequence number. We propose that violation of this unitary structure of palindromic sequences by means of our proposed symmetries leads to a mutation in DNA which could offer a new perspective in the scientific studies on the origin and cause of mutation.

A Sideways Look at Faithfulness for Quantum Correlations

Despite attempts to apply causal modeling techniques to quantum systems, Wood and Spekkens argue that any causal model purporting to explain quantum correlations must be fine tuned; it must violate the assumption of faithfulness. This paper is an attempt to undermine the reasonableness of the assumption of faithfulness in the quantum context. Employing a symmetry relation between an entangled quantum system and a “sideways” quantum system consisting of a single photon passing sequentially through two polarizers, I argue that Wood and Spekkens’s analysis applies equally to this sideways system also. As a result, we must either reject a causal explanation in this single photon system, or the sideways system must be fine tuned. If the latter, a violation of faithfulness in the ordinary entangled system may be more tolerable than first thought. Thus, extending the classical “no fine-tuning” principle of parsimony to the quantum realm may be too hasty.

Emergence of reflexivity relation without identity matching-to-sample training in hooded crows (Corvus cornix)

The ability to form equivalent relations between sign and referent—symbolization—is one of the important cognitive components of language. Equivalent relations have the properties of symmetry (if A→B then B→A), reflexivity (A→A, B→B), and transitivity (if A→B and B→C, then A→C). The current study evaluates whether reflexivity can be spontaneously revealed in hooded crows (Corvus cornix) without training after the formation of the symmetry relation. These birds were previously taught an arbitrary matching-to-sample task with the letters “S” and “V” as samples, and sets of images (same-sized and different-sized figures) as comparisons. Positive results in the transfer tests showed that the crows associated letters with the concepts of sameness/difference. After that, they successfully passed the symmetry test, in which samples and comparisons were switched around. In the present experiment we found out that the crows passed the reflexivity test (A→A, B→B) without identity training. We hypothesize that if the subject associates the sample not with certain stimuli but rather with concepts, it facilitates the formation of equivalence relations between them.

A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection

International audience We investigate the combinatorics of the symmetry relation H μ(x; q, t) = H μ∗ (x; t, q) on the transformed Macdonald polynomials, from the point of view of the combinatorial formula of Haglund, Haiman, and Loehr in terms of the inv and maj statistics on Young diagram fillings. By generalizing the Carlitz bijection on permutations, we provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) for the coefficients of the square-free monomials in the variables x. Our work in this case relates the Macdonald inv and maj statistics to the monomial basis of the modules Rμ studied by Garsia and Procesi. We also provide a new proof for the full Macdonald relation in the case when μ is a hook shape.

Fluctuation relations and fitness landscapes of growing cell populations

We construct a pathwise formulation of a growing population of cells, based on two different samplings of lineages within the population, namely the forward and backward samplings. We show that a general symmetry relation, called fluctuation relation relates these two samplings, independently of the model used to generate divisions and growth in the cell population. Known models of cell size control are studied with a formalism based on path integrals or on operators. We investigate some consequences of this fluctuation relation, which constrains the distributions of the number of cell divisions and leads to inequalities between the mean number of divisions and the doubling time of the population. We finally study the concept of fitness landscape, which quantifies the correlations between a phenotypic trait of interest and the number of divisions. We obtain explicit results when the trait is the age or the size, for age and size-controlled models.

SYMMETRY GEOMETRY BY PAIRINGS

In this paper, we introduce asymmetry geometryfor all those mathematical structures which can be characterized by means of a generalization (which we call pairing) of a finite rectangular table. In more detail, let$\unicode[STIX]{x1D6FA}$be a given set. Apairing$\mathfrak{P}$on$\unicode[STIX]{x1D6FA}$is a triple$\mathfrak{P}:=(U,F,\unicode[STIX]{x1D6EC})$, where$U$and$\unicode[STIX]{x1D6EC}$are nonempty sets and$F:U\times \unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6EC}$is a map having domain$U\times \unicode[STIX]{x1D6FA}$and codomain$\unicode[STIX]{x1D6EC}$. Through this notion, we introduce a local symmetry relation on$U$and a global symmetry relation on the power set${\mathcal{P}}(\unicode[STIX]{x1D6FA})$. Based on these two relations, we establish the basic properties of our symmetry geometry induced by$\mathfrak{P}$. The basic tool of our study is a closure operator$M_{\mathfrak{P}}$, by means of which (in the finite case) we can represent any closure operator. We relate the study of such a closure operator to several types of others set operators and set systems which refine the notion of an abstract simplicial complex.

The gauge action, DG Lie algebras and identities for Bernoulli numbers

In this paper we prove a family of identities for Bernoulli numbers parameterized by triples of integers ${(a,b,c)}$ with ${a+b+c=n-1}$, ${n\geq 4}$. These identities are deduced by translating into homotopical terms the gauge action on the Maurer–Cartan set of a differential graded Lie algebra. We show that Euler and Miki’s identities, well-known and apparently non-related formulas, are linear combinations of our family and they satisfy a particular symmetry relation.

A Combinatorial Approach to the $q,t$-Symmetry Relation in Macdonald Polynomials

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\widetilde{H}_\mu(\mathbf{x};q,t)=\widetilde{H}_{\mu^\ast}(\mathbf{x};t,q)$. We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials ($q=0$) when $\mu$ is a partition with at most three rows, and for the coefficients of the square-free monomials in $\mathbf{x}$ for all shapes $\mu$. We also provide a proof for the full relation in the case when $\mu$ is a hook shape, and for all shapes at the specialization $t=1$. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.

The Maaß space and Hecke operators

We give a new proof of the fact that the Maaß space is invariant under all Hecke operators. It is based on the characterization of the Maaß space by a symmetry relation and certain commutation relations of the Hecke algebra for the Jacobi group.