Narrow Results

In this paper, we introduce the notions of the directed, undirected, compressed directed, and compressed undirected irreducible divisor graphs $\mathrm{\Gamma }(x)$, $G(x)$, ${\mathrm{\Gamma }}_c(x),$ and $G_c(x)$ of a nonzero nonunit $x$ in a noncommutative atomic domain $D$, respectively. In light of these notions, we identify the corresponding characterizations of the normal unique factorization domain (n-UFD). Moreover, we study the connection between the kinds of directed and undirected irreducible divisor graphs and an n-FFD. Consequently, these characterizations generalize the results of Coykendall and Maney [Irreducible divisor graphs, Comm. Algebra35(3) (2007) 885895] and Axtell et al. [Irreducible divisor graphs and factorization properties of domains, Comm. Algebra39(11) (2011) 41484162] to the noncommutative setting.

The set of all terms of type $\tau$ is denoted by $W_{\tau }(X).$ Each element of $P(W_{\tau }(X))$ is called a tree language. A binary operation ${\cdot }_{ij}$ is an operation on $P(W_{\tau }(X))$ defined from the superposition ${Ŝ}_g^n$ on the set of all $n$-ary terms of type $\tau .$ Since ${Ŝ}_g^n$ satisfies the superposition law, $(P(W_{\tau }(X));{\cdot }_{ij})$ is a semigroup. In this paper, we study subsemigroups of the semigroup $(P(W_{\tau }(X));{\cdot }_{ij})$ including constant, left-zero, right-zero subsemigroups, rectangular bands, subgroups, and inverse subsemigroups. We also study factorization and locally factorization of the semigroup $(P(W_{\tau }(X));{\cdot }_{ij})$ and we obtain that this semigroup is neither factorizable nor locally factorizable. However, we achieve a necessary and sufficient condition for factorization of local subsemigroup $A{\cdot }_{ij}P(W_{\tau }(X)){\cdot }_{ij}A$, where $A$ is idempotent in $P(W_{\tau }(X))$. Finally, we study ${\cdot }_{ij}-$iteration and show that with special conditions, an ${\cdot }_{ij}-$iteration of idempotent is idempotent.

The paper presents results on factorization by similarity of fuzzy concept lattices with hedges. A fuzzy concept lattice is a hierarchically ordered collection of clusters extracted from tabular data. The basic idea of factorization by similarity is to have, instead of a possibly large original fuzzy concept lattice, its factor lattice. The factor lattice contains less clusters than the original concept lattice but, at the same time, represents a reasonable approximation of the original concept lattice and provides us with a granular view on the original concept lattice. The factor lattice results by factorization of the original fuzzy concept lattice by a similarity relation. The similarity relation is specified by a user by means of a single parameter, called a similarity threshold. Smaller similarity thresholds lead to smaller factor lattices, i.e. to more comprehensible but less accurate approximations of the original concept lattice. Therefore, factorization by similarity provides a trade-off between comprehensibility and precision.

We first describe the notion of factorization. Second, we present a way to compute the factor lattice directly from input data, i.e. without the need to compute the possibly large original concept lattice. Third, we provide an illustrative example to demonstrate our method.

We say a family of strings over an alphabet is an UMFF if every string has a unique maximal factorization over . Foundational work by Chen, Fox and Lyndon established properties of the Lyndon circ-UMFF, which is based on lexicographic ordering. Commencing with the circ-UMFF related to V-order, we then proved analogous factorization families for a further 32 Block-like binary orders. Here we distinguish between UMFFs and circ-UMFFs, and then study the structural properties of circ-UMFFs. These properties give rise to the complete construction of any circ-UMFF. We prove that any circ-UMFF is a totally ordered set and a factorization over it must be monotonic. We define atom words and initiate a study of u, v-atoms. Applications of circ-UMFFs arise in string algorithmics.

In [A. Frid, S. Puzynina and L. Q. Zamboni, On palindromic factorization of words, Adv. in Appl. Math.50 (2013) 737–748], it was conjectured that any infinite word whose palindromic lengths of factors are bounded is ultimately periodic. We introduce variants of this conjecture and prove this conjecture when the bound is 2. Especially we introduce left and right greedy palindromic lengths. These lengths are always greater than or equals to the initial palindromic length. When the greedy left (or right) palindromic lengths of prefixes of a word are bounded then this word is ultimately periodic.

Let $a,k\in \mathbb{N}$. By ${}^{1}a:=a$ and ${}^{k}a:=a^{{}^{k-1}a}$, we denote the $k$ th iterate of the exponential function $x\mapsto a^x$ evaluated at $a$, also known as tetration. We demonstrate how an algorithm for evaluating tetration modulo natural numbers $N$ could be used to compute the prime factorization of $N$ and provide heuristic arguments for the efficiency of this reduction. Additionally, we prove that the problem of computing the squarefree part of integers is deterministically polynomial-time reducible to modular tetration.

Two-jet event shape distributions, traditionally studied in the language of perturbative QCD, can be described naturally in soft-collinear effective theory. In this language, we demonstrate factorization of event shape distributions into perturbatively-calculable hard and jet functions and nonperturbative soft functions, and show how the latter contribute universal shifts to the mean values of various event shape distributions. Violations of universality in shifts of higher moments can give information on correlations of energy flow in soft radiation.

Centrality dependence of nuclear modification factor at high p_T above 4 GeV/c is studied in nucleus–nucleus collisions at . We find that the centrality dependence of the nuclear modification factor can be factorized as a Boltzmann function F(b). Comparing our model calculation with PHENIX data, we further confirm that the high p_T spectrum of particles is dominated by surface emission.

After the final analyses of the H1 and ZEUS collaborations for the diffractive photoproduction of dijets have appeared, we have recalculated these cross sections in next-to-leading order (NLO) of perturbative QCD to see whether they can be interpreted consistently. The results of these calculations are compared to the data of both collaborations. We find that at NLO the cross sections disagree with the data, showing that factorization breaking occurs at this order. If direct and resolved contributions are both suppressed by the same amount, the global suppression factor depends on the transverse-energy cut and is 0.42 for the H1 and 0.71 for the ZEUS analysis. However, by suppressing only the resolved contribution by a factor of approximately three, also reasonably good agreement with all the data is found. The size of the factorization breaking effects for resolved photons agrees with absorptive-model predictions.

Recent results on the application of the method of the effective quanta to the case of WW scattering are reviewed. In particular, it is reviewed how effective W bosons can be used to compute scattering amplitudes in a generalized Effective W Approximation (gEWA). The physical origin and the expected size of the discrepancy between the exact computation and the gEWA computation are discussed. Particular attention is devoted to the peculiarities of the formulation of the gEWA for processes that involve longitudinal W bosons. These formal results are checked through explicit numerical calculations of both scattering amplitudes and scattering cross-sections. Furthermore, the use of the gEWA for the interpretation and for the presentation of the measurement of WW scattering at the LHC is briefly discussed.

It is known that the barotropic FRW system of differential equations for zero cosmological constant can be reduced to simple harmonic oscillator (HO) differential equations in the conformal time variable. This is due to the fact that the Hubble rate parameter in conformal time is the solution of a simple Riccati equation of constant coefficients. In previous works, we have used this mathematical result to set the barotropic HO equations in the nonrelativistic supersymmetric approach by factorizing them. If a constant additive parameter, denoted by S, is added to the common Riccati solution of these supersymmetric partner cosmologies one obtains inhomogeneous barotropic cosmologies with periodic singularities in their spatial curvature indices that are counterparts of the non-shifted supersymmetric partners. The zero-mode solutions of these cyclic singular cosmologies are reviewed here as a function of real and imaginary shift parameter. We also notice the modulated zero modes obtained by using the general Riccati solution and comment on their cosmological application.

We report on recent progress in testing the factorization formalism of non-relativistic quantum chromodynamics (NRQCD) at next-to-leading order (NLO) for J/ψ yield and polarization. We demonstrate that it is possible to unambiguously determine the leading color-octet (CO) long-distance matrix elements (LDMEs) in compliance with the velocity scaling rules through a global fit to experimental data of unpolarized J/ψ production in pp, , ep, γγ, and e⁺e^- collisions. Three data sets not included in the fit, from hadro-production and from photo-production in the fixed-target and colliding-beam modes, are nicely reproduced. The polarization observables measured in different frames at DESY HERA and CERN LHC reasonably agree with NLO NRQCD predictions obtained using the LDMEs extracted from the global fit, while measurements at the FNAL Tevatron exhibit severe disagreement. We demonstrate that the alternative LDME sets recently obtained, with different philosophies, in two other NLO NRQCD analyses of J/ψ yield and polarization also fail to reconcile the Tevatron polarization data with the other available world data.

The dependence of inclusive one jet differential cross-section on the renormalization and factorization scale is studied. The choice when both scales μ_r, μ_f are set equal to jet transversal momentum is taken as a referential point and within a region around this choice, the inclusive one jet differential cross-section is calculated at next-to-leading order. Following the principle of minimal sensitivity (PMS), the region of local stability with respect to both scales is searched for. It is shown that it is well approximated by particular choice for total energies 7000 and 14,000 GeV.

In this work, we study the radiative leptonic decays of $B^{-}$ and $D^{-}$ mesons in the Standard Model (SM) and two-Higgs-doublet-model (2HDM) Type II. The results are obtained using the factorization procedure, and the contribution of the order $O({\mathrm{\Lambda }}_{\mathrm{\text{QCD}}}/m_Q)$ is included. The numerical results are calculated using the wave function obtained in relativistic potential model. As a result, the decay mode $B\to \gamma \tau {\nu }_{\tau }$ is found to be sensitive to the effect of the charged Higgs boson. Using the constraint on the free parameters of 2HDM given in previous works, we find the contribution of the charged Higgs boson in the decay mode $B\to \gamma \tau {\nu }_{\tau }$ can be as large as about 13%.

Nonleptonic $B_c$ decays to $P$-wave charmonia and light ${\pi }^{+}$, $K^{+}$, and ${\rho }^{+}$, $K^{\ast +}$ mesons are analyzed using factorization. The hadronic form factors parametrizing the $B_c\to {\chi }_{cJ}(h_c)$ matrix elements are expressed in terms of universal functions at the leading order of an expansion in the relative velocity of the heavy quarks in the $B_c$ rest-frame and in $1/m_Q$. Several ratios of branching fractions are evaluated, and when experimental information can be used, single branching fractions are presented. Both the $1P$ and $2P$ charmonia are considered. If the exotic candidate state ${\chi }_{c1}(3872)$ corresponds to ${\chi }_{c1}(2P)$, it should be produced in nonleptonic $B_c$ decays with predicted abundances with respect to the other states in the charmonium $2P$ spin four-plet.

The purpose of this paper is to show that the cross-section factorization relation σ_nn(s)/σ_γp(s) = σ_γp(s)/σ_γγ(s) is satisfied experimentally in the energy domain , where the σ's are total cross-sections and nn denotes the even portion of the pp and total cross-section. A convenient phenomenological parametrization for a global simultaneous fit to the pp, , γp and γγ total cross-section data together with the ρ-value data for pp and is provided by using real analytic amplitudes. Within experimental errors, we show that factorization is satisfied when we unfold the published γγ data which had averaged the cross-sections obtained by using the two different PHOJET and PYTHIA Monte Carlo results. Our analysis clearly favors the PHOJET results and suggests that the additive quark model, together with vector meson dominance, allows one to compute σ_γp(s) and σ_γγ(s) from σ_nn(s) with essentially no free parameters. The universal ρ-value predicted by our fit, i.e. ρ_nn = ρ_γp = ρ_γγ, is compared to the ρ-value obtained by a QCD-inspired analysis of and pp data, including the p-air cross-sections from cosmic rays. The ρ-values obtained from the two techniques are essentially indistinguishable in the energy region , giving us increased confidence in our parametrization of the cross-sections needed for the factorization relation.

Experimental results on soft and hard diffractive processes obtained by the CDF Collaboration in interactions are examined with emphasis on regularities that point to QCD aspects of hadronic diffraction. Data are interpreted in a phenomenological approach in which diffractive cross sections are related to the underlying inclusive parton distribution functions of the nucleon. In this approach, diffraction appears to be mediated by the exchange of low-x partons subject to color constraints.

Two-body nonleptonic charmless weak decays of Λ_b baryon involving p (or Λ) have been discussed. The form factors for various transitions are calculated in the nonrelativistic quark model, based on factorization assumption.

Based on IS model, the cross-sections of associated η_c and γ diffractive production in collision are calculated. Under the framework of NRQCD factorization formalism for quarkonia production, the dominant contribution to associated η_c and γ production comes from color octet subprocess, the color octet matrix element of is related to that of of J/ψ by the heavy quark spin symmetry. Because the value of color octet matrix element of of J/ψ is determined by the large P_TJ/ψ production data, the prediction of η_c and γ is more reliable. The ratio of diffractive to inclusive cross-sections is independent of the values of color octet matrix elements, which is sensitive to the gluon factor of the Pomeron and renormalized Pomeron flux factors. So experimental measurement of this ratio can give us more information of the nature of Pomeron and test the assumption of diffractive hard scattering factorization.

A brief summary of outstanding theoretical issues and recent results from the QCD factorization approach to exclusive hadronic B decays is provided.