Logarithmic Structure Research Articles

An energy-based framework for predicting vehicle noise source intensity: From energy consumption to noise

The vehicle noise source strength prediction model is a crucial component in the field of traffic noise prediction. Despite the establishment of noise source strength localized models in various countries, the theoretical underpinnings of the sound power level models within these frameworks remains unclear. This study addresses this gap by analyzing the correlation between vehicle noise and energy consumption. An energy-based source strength model framework (E-SSIM) is proposed, focusing on developing nonlinear models for basic noise level. E-SSIM is built on acoustical principles and the energy flow of vehicles, integrating noise and energy consumption through the application of multivariate regression theory, characterized by a transient or simplified mathematical framework. Furthermore, sensitivity analysis and road experiments are conducted to validate the proposed framework. The findings reveal that E-SSIM effectively integrates vehicle energy flow and principles of acoustics, thereby providing a theoretical foundation for the logarithmic mathematical structure in classical noise source strength models. The study reveals that in low-speed driving conditions (17–40 km/h), the sensitivity of noise energy to aerodynamic drag energy consumption reaches its peak. Specifically, the sensitivity of E-SSIM, as assessed by the A-weighted sound level, progressively decreases with increasing speed. On the contrary, for the Z-weighted sound level, the sensitivity initially decreases before rising again, reaching its peak stability and robustness at a speed of 23.8 km/h. E-SSIM exhibits superior precision in predicting A/Z-weighted sound pressure levels. Compared to classic logarithmic structural prediction models, the mean absolute percentage error of E-SSIM was reduced by 4.19% and 0.07%. Compared to typical models such as ASJ developed by the Acoustical Society of Japan and CNOSSOS-EU used by the European Commission, E-SSIM yielded a mean absolute percentage error reduction of 68% and 67%. Interestingly, as vehicle internal energy consumption increases, the prediction deviations of E-SSIM, ASJ, and CNOSSOS-EU gradually decrease, possibly because vehicle operating conditions approach stability. E-SSIM can utilize abundant vehicle data to develop generic models, promoting the advancement of noise prediction.

Period Integrals of Hypersurfaces via Tropical Geometry

Abstract Let $\left \{ Z_{t} \right \}_{t}$ be a one-parameter family of complex hypersurfaces of dimension $d \geq 1$ in a toric variety. We compute asymptotics of period integrals for $\left \{ Z_{t} \right \}_{t}$ by applying the method of Abouzaid–Ganatra–Iritani–Sheridan, which uses tropical geometry. As integrands, we consider Poincaré residues of meromorphic $(d+1)$-forms on the ambient toric variety, which have poles along the hypersurface $Z_{t}$. The cycles over which we integrate them are spheres and tori, which correspond to tropical $(0, d)$-cycles and $(d, 0)$-cycles on the tropicalization of $\left \{ Z_{t} \right \}_{t}$, respectively. In the case of $d=1$, we explicitly write down the polarized logarithmic Hodge structure of Kato–Usui at the limit as a corollary. Throughout this article, we impose the assumption that the tropicalization is dual to a unimodular triangulation of the Newton polytope.

Spatial Model of the Organization of Chromatin in the Nucleus of a Biological Cell According to Small-Angle Scattering Data

Small-angle neutron and X-ray scattering data for HeLa nuclei with the normal and suppressed transcription activities are reported. Small-angle neutron scattering data demonstrate the presence of a bifractal structure inside a nucleus. The logarithmic fractal structure is observed in the range from the size of the nucleus to several hundreds of nanometer, whereas the volume fractal structure exists at smaller scales down to a nucleosome structure. Small-angle X-ray scattering data show that the presence of the volume fractal structure correlates with the transcription activity of a cell. In view of the successful description of chromatin by the fractal globule model (Hi–C method data), a scenario for the formation of the bifractal structure inside the nucleus has been proposed. A system of transport channels (logarithmic fractal) is located inside close-packed chromatin, whereas active chromatin is localized near transport channels or inside them and forms volume fractal structures due to the transcription activity. Thus, the logarithmic fractal structure ensures the uniform distribution of voids at various scales, which is potentially necessary for the transcription and transport of substances inside the nucleus, whereas the volume fractal structure is due to the transcription activity of the cell.

Pointwise multipliers for Besov spaces Bp,∞0,b(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^{0,b}_{p,\\infty }({\\mathbb {R}}^n)$$\\end{document} with only logarithmic smoothness

In this article, we establish a characterization of the set M(Bp,∞0,b(Rn))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M(B^{0,b}_{p,\\infty }({\\mathbb {R}}^n))$$\\end{document} of all pointwise multipliers of Besov spaces Bp,∞0,b(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^{0,b}_{p,\\infty }({\\mathbb {R}}^n)$$\\end{document} with only logarithmic smoothness b∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b\\in {\\mathbb {R}}$$\\end{document} in the special cases p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document} and p=∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=\\infty $$\\end{document}. As applications of these two characterizations, we clarify whether or not the three concrete examples, namely characteristic functions of open sets, continuous functions defined by differences, and the functions eik·x\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^{ik\\cdot x}$$\\end{document} with k∈Zn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\in {\\mathbb {Z}}^n$$\\end{document} and x∈Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x\\in {\\mathbb {R}}^n$$\\end{document}, are pointwise multipliers of B1,∞0,b(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^{0,b}_{1,\\infty }({\\mathbb {R}}^n)$$\\end{document} and B∞,∞0,b(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^{0,b}_{\\infty ,\\infty }({\\mathbb {R}}^n)$$\\end{document}, respectively; furthermore, we obtain the explicit estimates of ‖eik·x‖M(B1,∞0,b(Rn))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert e^{ik \\cdot x}\\Vert _{M(B^{0,b}_{1,\\infty }({\\mathbb {R}}^n))}$$\\end{document} and ‖eik·x‖M(B∞,∞0,b(Rn))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert e^{ik \\cdot x}\\Vert _{M(B^{0,b}_{\\infty ,\\infty }({\\mathbb {R}}^n))}$$\\end{document}. In the case where p∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in (1,\\infty )$$\\end{document}, we give some sufficient conditions and some necessary conditions of the pointwise multipliers of Bp,∞0,b(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^{0,b}_{p,\\infty }({\\mathbb {R}}^n)$$\\end{document} and a complete characterization of M(Bp,∞0,b(Rn))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M(B^{0,b}_{p,\\infty }({\\mathbb {R}}^n))$$\\end{document} is still open. However, via a different method, we are still able to accurately calculate ‖eik·x‖M(Bp,∞0,b(Rn))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert e^{ik \\cdot x}\\Vert _{M(B^{0,b}_{p,\\infty }({\\mathbb {R}}^n))}$$\\end{document}, k∈Zn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\in {\\mathbb {Z}}^n$$\\end{document}, in this situation. The novelty of this article is that most of the proofs are constructive and these constructions strongly depend on the logarithmic structure of Besov spaces under consideration.

Time-of-flight spin-echo small-angle neutron scattering applied to biological cell nuclei

The organization of chromatin in the nuclei of rat lymphocyte was studied by time-of-flight spin-echo small-angle neutron scattering (ToF-SESANS). The procedures of the measurements in ToF and monochromatic SESANS modes were compared. It is shown that the sensitivity of the ToF method is significantly higher than that of the monochromatic mode. As a result, the SESANS correlation function can always be extracted from the ToF measurement of polarization, while this is not always the case with the monochromatic mode. The applicability of SESANS to fractal objects at the micrometre scale is demonstrated. The SESANS correlation function is well fitted by the exponential decay G(z) = exp(−z/ξ) with the correlation length ξ = 3.7 ± 0.1 µm. The exponential decay of the SESANS correlation function can be connected to the logarithmic correlation function γ(r) = ln(r/ξ) and the cubic law of the scattering intensity I(Q) = Q −3, which is in agreement with the concept of scattering from fractal objects. This finding is interpreted as evidence of a very specific logarithmic fractal structure of the large-scale organization of chromatin. The model of the logarithmic fractal is visualized as a hierarchical object obeying the volume-conservation principle at different scales.

Welding Penetration Monitoring for Ship Robotic GMAW Using Arc Sound Sensing Based on Improved Wavelet Denoising

The arc sound signal is one of the most important aspects of information related to pattern identification regarding the penetration state of ship robotic GMAW; however, arc sound is inevitably affected by noise interference during the signal acquisition process. In this paper, an improved wavelet threshold denoising method is proposed to eliminate interference and purify the arc sound signal. The non-stationary random distribution characteristics of GMAW noise interference are also estimated by using the high-frequency detail coefficients in different domains after wavelet transformation, and a mode of measuring scale that is logarithmically negatively correlated with the wavelet decomposition scale is created to update the threshold. The gradient convergent threshold function is established using the natural logarithmic function structure and concave–convex gradient to enable the nonlinear adjustment of the asymptotic rate. Further, some property theorems related to the optimized threshold function are proposed and theoretically proven, and the effectiveness and adaptability of the improved method are verified via the denoising simulation of speech synthesis signals. The four traditional denoising methods and our improved version are applied in the pretreatment of the GMAW arc sound signal, respectively. Statistical analysis and short-time Fourier transform are used to extract eight-dimensional time and frequency domain feature parameters from the denoised signals with randomly time-varying characteristics, and the extracted joint feature parameters are used to establish a nonlinear mapping model of penetration state identification for ship robotic GMAW using the pattern classifiers of RBFNN, PNN and PSO-SVM. The simulation results yielded by visual penetration classification and the multi-dimensional evaluation index of the confusion matrix indicate that the improved denoising method proposed in this paper achieves a higher accuracy in the extraction of penetration state features and greater precision in the identification of pattern classification.

On the stability of Yang-Mills vacuum

The (anti)self-dual covariantly constant vacuum fields have positive/stable and infinitely many zero modes, so called Leutwyller chromons. A regularisation method is suggested that allows to sum the contribution of the zero modes and to obtain the effective Lagrangian that has contribution of all positive/stable modes and zero modes. In the second approach we integrated over the collective coordinates of the nonlinearly interacting zero modes and obtained the same result. The last method is a generalisation of the t'Hooft integration over the instanton zero mode collective coordinates. It is important that both methods lead to the same result indicating the robustness of the logarithmic structure of the effective Lagrangian.

Dead-cone searches in heavy-ion collisions using the jet tree

We explore the possibility of using the dead cone of heavy quarks as a region of the Lund plane where medium-induced gluon radiation can be isolated and characterised. The filling of the dead cone by medium-induced gluons is expected to be the result of the interplay between the minimum angle of such radiation due to transverse momentum broadening and the dead-cone angle. Since the measurement of a fully corrected Lund plane in heavy-ion collisions is currently challenging, we propose to use jet grooming techniques to identify a particular splitting in the jet tree that is both perturbative and sensitive to the dead-cone effect. To that end, we propose a new jet substructure groomer, dubbed Late-$k_t$, that selects the most collinear splitting in a QCD jet above a certain transverse momentum cutoff $k_{t,\rm{cut}}$. The role of $k_{t,\rm{cut}}$ is to guarantee perturbative splittings, while selecting the most collinear splitting enhances the sensitivity to mass effects. As a proof of concept, we study the angular distribution of the splitting tagged by Late-$k_t$ both analytically and with Monte Carlo simulations. First, we derive the logarithmic resummation structure in vacuum and demonstrate its capability to distinguish between inclusive and heavy-flavoured jets. Next, we extend the calculation for in-medium jets and show that medium-induced emissions lead to an enhancement of collinear emissions below the dead cone angle. Numerically, we demonstrate an excellent resilience of Late-$k_t$ against uncorrelated thermal background, thus confirming this observable as a potential candidate to unveil medium dynamics around the dead cone regime.

Analytic semi-universal deformations in logarithmic complex geometry

We show that every compact complex analytic space endowed with a fine logarithmic structure and every morphism between such spaces admit a semi-universal deformation. These results generalize the analogous results in complex analytic geometry first independently proved by A. Douady and H. Grauert in the ’70. We follow Douady’s two steps process approach consisting of an infinite-dimensional construction of the deformation space followed by a finite-dimensional reduction.

A logarithmic sliding mode controller for stochastic active suspension systems

In this paper, a new approach for stochastic logarithmic variable structure control has been proposed to create a compromise between reaching phase duration and response rate of the stochastic suspension system. The asymptotic stability with probability one for the closed-loop suspension system has been analyzed by Lyapunov method. Some practical considerations in proving theorems are uncertain parameter variations, environmental matched disturbances, actuator degradation, stochastic environmental mismatched disturbances, which is a representative of the change in road profile with time, and unmodeled dynamics. Finally, by regulating a specific parameter, the simulation results corroborate the advantage of the proposed controller in managing and reducing the sensitivity of the closed-loop system. Also, compared with other sliding surfaces of past researches, the designed sliding variable minimizes a certain performance function and therefore provides better performance for the closed-loop system, which is also verified in the simulation results.

OPTIMAL TEMPERATURE DISTRIBUTION FOR A NONISOTHERMAL CAHN-HILLIARD SYSTEM IN TWO DIMENSIONS WITH SOURCE TERM AND DOUBLE OBSTACLE POTENTIAL

In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the con­stitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails further mathematical difficulties because the mass conservation of the order parameter is no longer satisfied. In this paper, we study the case that the double-well potential driving the evolution of the phase transition is given by the nondifferentiable dou­ble obstacle potential, thereby complementing recent results obtained for the differentiable cases of regular and logarithmic potentials. Be­sides existence results, we derive first-order necessary optimality condi­tions for the control problem. The analysis is carried out by employing the so-called deep quench approximation in which the nondifferentiable double obstacle potential is approximated by a family of potentials of logarithmic structure for which meaningful first-order necessary opti­mality conditions in terms of suitable adjoint systems and variational inequalities are available. Since the results for the logarithmic poten­tials crucially depend on the validity of the so-called strict separation property which is only available in the spatially two-dimensional situ­ation, our whole analysis is restricted to the two-dimensional case.

Water use efficiency evolution in the Yellow River Basin: an integrated analysis of spatial-temporal decomposition

ABSTRACT Understanding water use efficiency evolution and its driving factors should underpin water use efficiency improvement and water demand management. Here I present an integrated analysis framework combining spatial and temporal decomposition analyses to assess sub-basin water use efficiency performances in the Yellow River Basin. Two well-established decomposition approaches, i.e. logarithmic mean Divisia index decomposition and structure decomposition analysis, are used to break down spatial and temporal changes of water use efficiency variables into different contributing factors, and the sensitivity of the result to decomposition methods is examined. The results indicate that sub-basin water use efficiencies overall tend to improve, but show large variability in the magnitude of improvement. The graph of pathways provides a clear virtual presentation to understand the evolution of contributions from different socio-economic factors to water use efficiency changes. The results help in understanding water use changes and have important implications for sustainable water use and management.

Chaos of the Six-Dimensional Non-Autonomous System for the Circular Mesh Antenna

In the process of aerospace service, circular mesh antennas generate large nonlinear vibrations under an alternating thermal load. In this paper, the Smale horseshoe and Shilnikov-type multi-pulse chaotic motions of the six-dimensional non-autonomous system for circular mesh antennas are first investigated. The Poincare map is generalized and applied to the six-dimensional non-autonomous system to analyze the existence of Smale horseshoe chaos. Based on the topological horseshoe theory, the three-dimensional solid torus structure is mapped into a logarithmic spiral structure, and the original structure appears to expand in two directions and contract in one direction. There exists chaos in the sense of a Smale horseshoe. The nonlinear equations of the circular mesh antenna under the conditions of the unperturbed and perturbed situations are analyzed, respectively. For the perturbation analysis of the six-dimensional non-autonomous system, the energy difference function is calculated. The transverse zero point of the energy difference function satisfies the non-degenerate conditions, which indicates that the system exists Shilnikov-type multi-pulse chaotic motions. In summary, the researches have verified the existence of chaotic motion in the six-dimensional non-autonomous system for the circular mesh antenna.

Thermo‐mechanical nonlinear stability analysis of geometrically imperfect functionally graded plates with microstructural defects using logarithmic structure kinematics: An unified expression

AbstractIn the present paper, thermo‐mechanical stability analysis of geometrically imperfect porous functionally graded plates (FGP) with geometric nonlinearity is presented. The equilibrium, stability, and compatibility equations are derived using Logarithmic structural kinematics in conjunction with the von‐Karman type of geometric nonlinearity. A logarithmic‐based shear‐strain function has been used for the first time for the buckling response of functionally graded material (FGM) plates. Generic imperfection function has been implemented to incorporate the various imperfection modes like sin‐type or global type in the formulation. The effective materials properties of the plates with porosity inclusion have been computed using modified power law. The plate is subjected to uniaxial and biaxial compression as well as combined compression and tension along with thermal loading. An exact expression for the critical buckling load and critical buckling thermal load of geometrically imperfect porous FGM plate for each loading case has been developed. After confirming the excellent accuracy of the current exact solutions, the effect of geometric imperfection, porosity inclusion, and geometric configurations on the nonlinear stability of the FGM plate have been discussed extensively. The results presented in this paper will be used as a benchmark for future research.

On the logarithmic coarse structures of Lie groups and hyperbolic spaces

We characterize the Lie groups with finitely many connected components that are O(u)-bilipschitz equivalent (almost quasiisometric in the sense that the sublinear function u replaces the additive bounds of quasiisometry) to the real hyperbolic space, or to the complex hyperbolic plane. The characterizations are expressed in terms of deformations of Lie algebras and in terms of pinching of sectional curvature of left-invariant Riemannian metrics in the real case. We also compare sublinear bilipschitz equivalence and coarse equivalence, and prove that every coarse equivalence between the logarithmic coarse structures of geodesic spaces is a \(O(\log )\)-bilipschitz equivalence. The Lie groups characterized are exactly those whose logarithmic coarse structure is equivalent to that of a real hyperbolic space or the complex hyperbolic plane. Finally we point out that a conjecture made by Tyson about the conformal dimensions of the boundaries of certain hyperbolic buildings holds conditionally to the four exponentials conjecture.

Gromov–Witten theory and invariants of matroids

We use techniques from Gromov–Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov–Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement’s wonderful model is a combinatorial invariant, i.e., it depends only on the matroid. When the boundary divisor is maximal, we use toric intersection theory to convert the virtual fundamental class into a balanced weighted fan in a vector space, having the expected dimension. We explain how the associated Gromov–Witten theory is completely encoded by intersections with this weighted fan. We include a number of questions whose positive answers would lead to a well-defined Gromov–Witten theory of non-realizable matroids.

Next to soft corrections to Drell-Yan and Higgs boson productions

We present a framework that resums threshold enhanced large logarithms to all orders in perturbation theory for the production of a pair of leptons in Drell-Yan process and of Higgs boson in gluon fusion as well as in bottom quark annihilation. We restrict ourselves to contributions from diagonal partonic channels. These logarithms include the distributions $((1-z)^{-1} \ln^i(1-z))_+$ resulting from soft plus virtual (SV) and the logarithms $\ln^i(1-z)$ from next-to-SV (NSV) contributions. We use collinear factorisation and renormalisation group invariance to achieve this. The former allows one to define a Soft-Collinear (SC) function which encapsulates soft and collinear dynamics of the perturbative results to all orders in strong coupling constant. The logarithmic structure of these results are governed by universal infrared anomalous dimensions and process dependent functions of Sudakov differential equation that the SC satisfies. The solution to the differential equation is obtained by proposing an all-order ansatz in dimensional regularisation, owing to several state-of-the-art perturbative results available to third order. The $z$ space solutions thus obtained provide an integral representation to sum up large logarithms originating from both soft and collinear configurations, conveniently in Mellin $N$ space. We show that in $N$ space, tower of logarithms $a_s^n/N^\alpha \ln^{2n-\alpha} (N), a_s^n/N^\alpha \ln^{2n-1-\alpha}(N) \cdots $ etc for $\alpha =0,1$ are summed to all orders in $a_s$.

Experimental evidence for logarithmic fractal structure of botanical trees.

The area-preserving rule for botanical trees by Leonardo da Vinci is discussed in terms of a very specific fractal structure, a logarithmic fractal. We use a method of the numerical Fourier analysis to distinguish the logarithmic fractal properties of the two-dimensional objects and apply it to study the branching system of real trees through its projection on the two-dimensional space, i.e., using their photographs. For different species of trees (birch and oak) we observe the Q^{-2} decay of the spectral intensity characterizing the branching structure that is associated with the logarithmic fractal structure in two-dimensional space. The experiments dealing with the side view of the tree should complement the area preserving Leonardo's rule with one applying to the product of diameter d and length l of the k branches: d_{i}l_{i}=kd_{i+1}l_{i+1}. If both rules are valid, then the branch's length of the next generation is sqrt[k] times shorter than previous one: l_{i}=sqrt[k]l_{i+1}. Moreover, the volume (mass) of all branches of the next generation is a factor of d_{i}/d_{i+1} smaller than previous one. We conclude that a tree as a three-dimensional object is not a logarithmic fractal, although its projection onto a two-dimensional plane is. Consequently, the life of a tree flows according to the laws of conservation of area in two-dimensional space, as if the tree were a two-dimensional object.

Bifractal structure of chromatin in rat lymphocyte nuclei.

The small-angle neutron scattering (SANS) on the rat lymphocyte nuclei demonstrates the bifractal nature of the chromatin structural organization. The scattering intensity from rat lymphocyte nuclei is described by power law Q^{-D} with fractal dimension approximately 2.3 on smaller scales and 3 on larger scales. The crossover between two fractal structures is detected at momentum transfer near 10^{-1}nm^{-1}. The use of contrast variation (D_{2}O-H_{2}O) in SANS measurements reveals clear similarity in the structural organizations of nucleic acids (NA) and proteins. Both chromatin components show bifractal behavior with logarithmic fractal structure on the large scale and volume fractal with slightly smaller than 2.5 structure on the small scale. Scattering intensities from chromatin, protein component, and NA component demonstrate an extremely extensive range of logarithmic fractal behavior (from 10^{-3} to approximately 10^{-1}nm^{-1}). We compare the fractal arrangement of rat lymphocyte nuclei with that of chicken erythrocytes and the immortal HeLa cell line. We conclude that the bifractal nature of the chromatin arrangement is inherent in the nuclei of all these cells. The details of the fractal arrangement-its range and correlation/interaction between nuclear acids and proteins are specific for different cells and is related to their functionality.

Periods of tropical Calabi–Yau hypersurfaces

We consider the residual B-model variation of Hodge structure of Iritani defined by a family of toric Calabi–Yau hypersurfaces over a punctured disk D ∖ { 0 } D \setminus \left \{ 0\right \} . It is naturally extended to a logarithmic variation of polarized Hodge structure of Kato–Usui on the whole disk D D . By restricting it to the origin, we obtain a polarized logarithmic Hodge structure (PLH) on the standard log point. In this paper, we describe the PLH in terms of the integral affine structure of the dual intersection complex of the toric degeneration in the Gross–Siebert program.