Nonzero Number Research Articles

Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine

Abstract In this article, the authors introduce Qi’s normalized remainder of the Maclaurin series expansion of Qi’s normalized remainder for the cosine function. By virtue of a monotonicity rule for the quotient of two series and with the aid of an increasing monotonicity of a sequence involving the quotient of two consecutive non-zero Bernoulli numbers, they prove the logarithmic convexity of Qi’s normalized remainder. In view of a higher order derivative formula for the quotient of two functions, they expand the logarithm of Qi’s normalized remainder into a Maclaurin series whose coefficients are expressed in terms of determinants of a class of specific Hessenberg matrices. In light of a monotonicity rule for the quotient of two series, they present the monotonicity of the ratio between two normalized remainders. Finally, the authors connect two of their main results with the generalized hypergeometric functions.

Scalar tidal response of a rotating BTZ black hole

We study the response of a rotating BTZ black hole to the scalar tidal perturbation. We show that the real component of the tidal response function isn’t zero, indicating that a rotating BTZ black hole possesses non-zero tidal Love numbers. Additionally, we observe scale-dependent behaviour, known as log-running, in the tidal response function. We also conduct a separate analysis on an extremal rotating BTZ black hole, finding qualitative similarities with its non-extremal counterpart. In addition, we present a procedure to calculate the tidal response function of a charged rotating BTZ black hole as well.

Nonthermal Baryogenesis from Minimal Supersymmetric Standard Model Flat Direction

Abstract We study an inflection point inflation scenario where a flat direction of the minimal supersymmetric standard model (MSSM) is identified with the inflaton. We focus on the case where the flat direction (inflaton) has nonzero baryon number, and consider a nonthermal baryogenesis scenario where the decay of the inflaton at the reheating directly generates baryon asymmetry of the universe. Specifically, we consider a udd flat direction that is lifted by a superpotential operator of dimension 6, and show that inflection point inflation with the udd flat direction can be compatible with cosmological observations and can account for the baryon asymmetry of the universe.

Worst-case analysis of restarted primal-dual hybrid gradient on totally unimodular linear programs

We analyze restarted PDHG on totally unimodular linear programs. In particular, we show that restarted PDHG finds an ϵ-optimal solution in O(Hm12.5nnz(A)log⁡(Hm2/ϵ)) matrix-vector multiplies where m1 is the number of constraints, m2 the number of variables, nnz(A) is the number of nonzeros in the constraint matrix, H is the largest absolute coefficient in the right hand side or objective vector, and ϵ is the distance to optimality of the outputted solution.

ADJUNCTION OF ROOTS TO PROFINITE GROUPS

In this paper, when $G'$ is a group obtained by adjoining a $n$th-root of $g$ to a given group $G$, where $n$ is a nonzero natural number and $g$ is an element of $G$ of infinite order, we compute the profinite completion $\widehat{G'}$ of $G'$. Also, given $G$ a profinite group in which any subgroup of finite index is open, $n$ a nonzero natural number, $g$ an element of $G$, and $x$ an element not belonging to $G$, we point out necessary and sufficient condition under which the group obtained by adjoining roots to the profinite group $G$ remains again profinite. Our proofs make use of theoretico-combinatorial methods.

Inverse Element for Surreal Number

Summary Conway’s surreal numbers have a fascinating algebraic structure, which we try to formalise in the Mizar system. In this article, building on our previous work establishing that the surreal numbers fulfil the ring properties, we construct the inverse element for any non-zero number. For that purpose, we formalise the definition of the inverse element formulated in Section Properties of Division of Conway’s book. In this way we show formally in the Mizar system that surreal numbers satisfy all nine properties of a field.

A Note on Bi-Periodic Leonardo Sequence

In this work, we define a new generalization of the Leonardo sequence by the recurrence relation $GLe_n=aGLe_{n-1}+GLe_{n-2}+a$ (for even $n$) and $GLe_n=bGLe_{n-1}+GLe_{n-2}+b$ (for odd $n$) with the initial conditions $GLe_0=2a-1$ and $GLe_1=2ab-1$, where $a$ and $b$ are real nonzero numbers. Some algebraic properties of the sequence $\{GLe_n\}_{n \geq 0}$ are studied and several identities, including the generating function and Binet's formula, are established.

Half-quantized Hall states with C=3/2 exited by chiral photons

The topology of Hilbert space in semiconductor quantum wells (QW) can become non-trivial when excited by an electromagnetic wave. It means the quasi-energy bands acquire a nonzero integer Floquet-Chern number even if QW is trivial in the equilibrium. We analyze electron-photon interaction in the Floquet state and identify dynamical quantum phases characterized by half-quantized Floquet-Chern numbers. The non-integer topology depends on which equilibrium state is exposed to illumination. Trivial III-V semiconductor QW's such as AlGaAs and AlGaN, being resonantly illuminated, generate the dynamical index |CF|=1/2, similar to a gapless quantum anomalous semimetal (QAS) in equilibrium. Excitation of 2D topological insulator CdHgTe reveals |CF|=3/2 - the index stemmed exclusively from interaction with chiral photons. Comparing results obtained in continuous and full-Brillouin zone (BZ) models, we conclude that high-momentum regions in BZ contribute to the dynamical half-quantized topology. Results indicate that by manipulating parameters of external force, one can bring inverted-gap QW into a non-equilibrium state with a high topological non-integer charge |CF|=3/2. We present the phase diagram, which includes all dynamical states of non-trivial topology.

Dynamics and active mixing of a droplet in a Stokes trap

Particle trapping and manipulation have a wide range of applications in biotechnology and engineering. Recently, a flow-based, particle-trapping device called the Stokes trap was developed for trapping and control of small particles in the intersection of multiple branches in a microfluidic channel. This device can also be used to perform rheological experiments to determine the viscoelastic response of an emulsion or suspension. We show that besides these applications, the various flow modes produced by the Stokes trap are able to manipulate drop shapes and induce active mixing inside droplets. To this end, we analyse the dynamics of a droplet in a Stokes trap through boundary-integral simulations. We also explore the dynamic response of drop shape with respect to distinct external flow modes, which allows us to perform numerical experiments such as step strain and oscillatory extension. A linear controller is used to manipulate drop position, and the drop deformation is characterized by a spherical-harmonic decomposition. For small drop deformations, we observe a linear superposition of harmonics, which, surprisingly, seems to hold even for moderate deformations. This result indicates that such a device can be used for shape control of droplets. We also investigate how the different flow modes may be combined to induce mixing inside the droplets. The transient combination of modes produces an effective chaotic mixing, which is characterized by a mixing number. The mixing inside the droplet can be further enhanced for lower viscosity ratios and low, but non-zero capillary number and flow frequencies.

Exploring Seiberg-like N-alities with eight supercharges

We show that a large subclass of 3d 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 quiver gauge theories consisting of unitary and special unitary gauge nodes with only fundamental/bifundamental matter have multiple Seiberg-like IR duals. A generic quiver T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document} in this subclass has a non-zero number of balanced special unitary gauge nodes and it is a good theory in the Gaiotto-Witten sense. We refer to this phenomenon as IR N-ality and the set of mutually IR dual theories as the N-al set associated with the quiver T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document}. Starting from T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document}, we construct a sequence of dualities by step-wise implementing a set of quiver mutations which act locally on the gauge nodes. The associated N-al theories can then be read off from this duality sequence. The quiver T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document} generically has an emergent Coulomb branch global symmetry in the IR, such that the rank of the IR symmetry is always greater than the rank visible in the UV. We show that there exists at least one theory in the N-al set for which the rank of the IR symmetry precisely matches the rank of the UV symmetry. In certain special cases that we discuss in this work, the correct emergent symmetry algebra itself may be read off from this preferred theory (or theories) in addition to the correct rank. Finally, we give a recipe for constructing the 3d mirror associated with a given N-al set and show how the emergent Coulomb branch symmetry of T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{T} $$\\end{document} is realized as a UV-manifest Higgs branch symmetry of the 3d mirror. This paper is the second in a series of four papers on 3d 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 Seiberg-like dualities, preceded by the work [1].

Derivation of extended-OBurnett and super-OBurnett equations and their analytical solution to plane Poiseuille flow at non-zero Knudsen number

In this work, we analyse wall-bounded flows in the continuum to transition regime with the help of higher-order transport equations (super-set of the Navier–Stokes equation). Towards this, we incorporate second-order in Knudsen number accurate terms in the single-particle distribution function, and with this complete representation, we first derive the second-order accurate extended-OBurnett (EOBurnett) and third-order accurate super-OBurnett (SOBurnett) equations. We then demonstrate that these newly derived equations exhibit unconditional linear stability. We finally validate the equations by solving for plane Poiseuille flow and derive closed-form analytical solutions for the pressure and velocity fields. The pressure and velocity results thus obtained have been compared with direct simulation Monte Carlo (DSMC) data in the transition regime. The results from both the EOBurnett and SOBurnett equations are found to yield better agreement with DSMC data than that obtained from the Navier–Stokes equations. This improved agreement is attributed to the presence of additional terms in the proposed equations, which effectively capture the effect of the Knudsen layer near the wall. The obtained higher-order transport equations and the closed-form solution presented in this work are novel. The ability of the equations to describe the flow in the transition regime should form the basis for conducting further realistic analytical studies of wall-bounded flows in the future.

Structure of liquid-vapor interfaces: Perspectives from liquid state theory, large-scale simulations, and potential grazing-incidence x-ray diffraction.

Grazing-incidence x-ray diffraction (GIXRD) is a scattering technique that allows one to characterize the structure of fluid interfaces down to the molecular scale, including the measurement of surface tension and interface roughness. However, the corresponding standard data analysis at nonzero wave numbers has been criticized as to be inconclusive because the scattering intensity is polluted by the unavoidable scattering from the bulk. Here, we overcome this ambiguity by proposing a physically consistent model of the bulk contribution based on a minimal set of assumptions of experimental relevance. To this end, we derive an explicit integral expression for the background scattering, which can be determined numerically from the static structure factors of the coexisting bulk phases as independent input. Concerning the interpretation of GIXRD data inferred from computer simulations, we extend the model to account also for the finite sizes of the bulk phases, which are unavoidable in simulations. The corresponding leading-order correction beyond the dominant contribution to the scattered intensity is revealed by asymptotic analysis, which is characterized by the competition between the linear system size and the x-ray penetration depth in the case of simulations. Specifically, we have calculated the expected GIXRD intensity for scattering at the planar liquid-vapor interface of Lennard-Jones fluids with truncated pair interactions via extensive, high-precision computer simulations. The reported data cover interfacial and bulk properties of fluid states along the whole liquid-vapor coexistence line. A sensitivity analysis shows that our findings are robust with respect to the detailed definition of the mean interface position. We conclude that previous claims of an enhanced surface tension at mesoscopic scales are amenable to unambiguous tests via scattering experiments.

Novel study of inertial forces on MHD peristaltically driven micropolar fluid through porous-saturated asymmetric channel: Finite Galerkin approach

This focused study investigates the peristaltic motion of a micropolar fluid within an uneven channel filled with a porous medium, incorporating an orthogonal magnetic field to the flow. This research diverges from the traditional assumptions of lubrication theory. The governing equations, encompassing the physical characteristics of micropolar fluid peristalsis, are transformed into nonlinear coupled partial differential equations. These equations are solved using the finite element method, considering inertial effects, such as non-zero wave and Reynolds numbers. This study delves into the influence of various crucial parameters on axial velocity, pressure gradient, microrotation, and stream function, presenting graphical representations. Notably, the incremental phase shift causes an intermingling of upper and lower streamlines within both halves of the channel. As the Reynolds number increases, there is an observed reduction in bolus size, particularly at maximum phase shifts, with a tendency to move toward the central region. An increase in Hartmann number leads the bolus formation to vanish in both channels, reduces microrotation, and leads to increased pressure. Vorticity lines intensify and incline toward the peristaltic walls. An increase in the permeability parameter amplifies velocity, microrotation, volume, and bolus formation regardless of phase differences while countering pressure elevation per wavelength. Reduced concavity is observed as vorticity lines disperse across the entire area.

Investigations in Patterns in Last Digits of Square Numbers and Higher Powers

It is interesting to see what patterns exist in the final digits of powers of integers; the goal of this work is to introduce some new problems and results, which are ideally suited for interested readers to pursue further. We start our investigation with squares and say a square is a k-square good number if its last k digits base 10 are the same non-zero number, and the last k + 1 digits are not identical. We completely analyze k-square good numbers; when k = 1 we can have final digits of 1, 4, 5, 6 and 9 (with the second to last digit different), when k = 2 we can end with 44 but not 444, when k = 3 we can end with 444 but not 4444, and there are no 4-square good numbers. We then generalize these arguments to look at k-cube and k-fourth good numbers, completely analyzing these cases. Noting that any even power 2m is the same as squaring an mth power, we can extend our results to even powers more easily than to odd, as the behavior has to be a subset of what we have seen for squares. There are complications arising from powers of 2 dividing our numbers, but using results from elementary number theory (in particular concerning the totient function and Euler’s theorem), we reduce the problem for even powers to a tractable finite computation, and completely resolve this case, and then end with a list of accessible next problems.

Robust second-order topological insulator in 2D van der Waals magnet CrI3.

CrI3 offers an intriguing platform for exploring fundamental physics and the innovative design of spintronics devices in two-dimensional (2D) magnets, and moreover has been instrumental in the study of topological physics. However, the 2D CrI3 monolayer and bilayers have long been thought to be topologically trivial. Here we uncover a hidden facet of the band topology of 2D CrI3 by showing that both the CrI3 monolayer and bilayers are second-order topological insulators (SOTIs) with a nonzero second Stiefel-Whitney number w2 = 1. Furthermore, the topologically nontrivial nature can be explicitly confirmed via the emergence of floating edge states and in-gap corner states. Remarkably, in contrast to most known magnetic topological states, we put forward that the SOTIs in 2D CrI3 monolayer and bilayers are highly robust against magnetic transitions, which remain intact under both ferromagnetic and antiferromagnetic configurations. These interesting predictions not only provide a comprehensive understanding of the band topology of 2D CrI3 but also offer a favorable platform to realize magnetic SOTIs for spintronics applications.

Uhlmann phases in the Kitaev chain with NNN hopping interaction

We study the extended Kitaev chain with both nearest-neighbor and next-nearest-neighbor hopping terms and find the model system exhibiting nontrivial phases, which can be characterized by a nonzero Berry phase and winding number when the system is in a pure state. While in a mixed state, we investigate the robustness of the topological Uhlmann phases and show how it responses to the presence of next-nearest-neighbor hopping terms. Furthermore, we analyse the complicated behavior of the Uhlmann phase of the extended Kitaev chain at finite temperature as k moves along the Brillouin zone, and we think this may serve as a topological indicator for mixed states in condensed matter systems.

Filtered rays over iterated absolute differences on layers of integers

The dynamical system generated by the iterated calculation of the high order gaps between neighboring terms of a sequence of natural numbers is remarkable and only incidentally characterized at the boundary by the notable Proth–Gilbreath Conjecture for prime numbers.We introduce a natural extension of the original triangular arrangement, obtaining a growing hexagonal covering of the plane. This is just the base level of what further becomes an endless discrete helicoidal surface. Although the repeated calculation of higher-order gaps causes the numbers that generate the helicoidal surface to decrease, there is no guarantee, and most often it does not even happen, that the levels of the helicoid have any regularity, at least at the bottom levels.However, we prove that there exists a large and nontrivial class of sequences with the property that their helicoids have all levels coinciding with their base levels. This class includes in particular many ultimately binary sequences with a special header For almost all of these sequences, we additionally show that although the patterns generated by them seem to fall somewhere between ordered and disordered, exhibiting fractal-like and random qualities at the same time, the distribution of zero and non-zero numbers at the base level has uniformity characteristics.Thus, we prove that a multitude of straight lines that traverse the patterns encounter zero and non-zero numbers in almost equal proportions.

Effects of rotation on the rolling noise radiated by wheelsets in high-speed railways

Railway rolling noise is produced by the vibration of both the wheelsets and the track; the wheelsets dominate the high frequency noise and become increasingly important at high speed. The effect of rotation on the wheelset vibration and noise radiation is investigated using different wheelset models in a rolling noise prediction model for a wide range of speeds. Each wheelset model takes account of the rotation to a different extent. An axisymmetric finite element model of a flexible rotating wheelset is implemented based on a complex exponential formulation and expressed in either an inertial or a non-inertial frame of reference. The model can include the inertial Coriolis and centrifugal forces and is also extended to include stress stiffening. Modes of the rotating wheel with non-zero number of nodal diameters are split into co- and counter-rotating waves with separated natural frequencies. The extent of the frequency separation depends on the shape of the mode and its dominant component of vibration. At common train speeds the frequency shifts due to stress-stiffening and spin-softening effects are found to be small compared with the gyroscopic effects due to the Coriolis forces. The effect of including the inertial Coriolis and centrifugal forces on the overall A-weighted sound power level is less than 0.3 dB below 400 km/h, while for higher train speeds, differences may exceed 1 dB in some one-third octave bands. Overall, these differences are small compared with other sources of uncertainty in rolling noise modelling, confirming that representing the wheel rotation with a moving load approach provides a suitable approximation for use in rolling noise predictions.

An algebraic approach for counting DP-3-colorings of sparse graphs

DP-coloring (or correspondence coloring) is a generalization of list coloring that has been widely studied since its introduction by Dvořák and Postle in 2015. As the analogue of the chromatic polynomial of a graph G, P(G,m), and the list color function, Pℓ(G,m), the DP-color function of G, denoted by PDP(G,m), counts the minimum number of DP-colorings over all possible m-fold covers. It follows that PDP(G,m)≤Pℓ(G,m)≤P(G,m). A function f is chromatic-adherent if for every graph G, f(G,a)=P(G,a) for some a≥χ(G) implies that f(G,m)=P(G,m) for all m≥a. It is known that the DP-color function is not chromatic-adherent, but there are only two known graphs that demonstrate this. Suppose G is an n-vertex graph and H is a 3-fold cover of G. In this paper we associate with H a polynomial fG,H∈F3[x1,…,xn] so that the number of non-zeros of fG,H equals the number of H-colorings of G. We then use a well-known result of Alon and Füredi on the number of non-zeros of a polynomial to establish a non-trivial lower bound on PDP(G,3) when 2n>|E(G)|. An easy consequence of this is that PDP(G,3)≥3n/6 for every n-vertex planar graph G of girth at least 5, improving the previously known bounds on both PDP(G,3) and Pℓ(G,3). Finally, we use this bound to show that there are infinitely many graphs that demonstrate the non-chromatic-adherence of the DP-color function.

The reciprocating and bipolar non-Hermitian skin effect engineered by spin–orbit coupling

We theoretically study a one-dimensional non-Hermitian Su–Schrieffer–Heeger model with an imaginary gauge field and spin–orbit coupling. We find that, under open boundary conditions, the dispersions possess the reciprocating real–complex–real transitions with increasing the strength of spin–orbit coupling. Correspondingly, the bulk energy eigenstates exhibit a reciprocating non-Hermitian skin effect. This mechanism can be characterized by the generalized Brillouin zone moduli, which approaches zero or infinity at the transition points. We further demonstrate the non-zero winding number inside the loops of generalized Brillouin zone when the strength of intra-cell spin–orbit coupling is larger than the inter-cell one, which results in the bipolar non-Hermitian skin effect. As the intra-cell and inter-cell spin–orbit coupling strength becomes comparable, the bipolar non-Hermitian skin effect degenerates to the conventional non-Hermitian skin effect. Our work may pave the way for the non-Hermitian optoelectronic devices utilizing the reciprocating non-Hermitian skin effect and the bipolar non-Hermitian skin effect by engineering the spin–orbit coupling.