Small Positive Constant Research Articles

Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: Beyond the Lions exponent

We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent α can be larger than the Lions exponent 5/4. It is well-known that, due to Lions [1], for any L2 divergence-free initial data, there exist unique smooth Leray-Hopf solutions when α≥5/4. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces LtγWxs,p, in view of the Ladyženskaja-Prodi-Serrin criteria. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints (3/p+1−2α,∞,p) and (2α/γ+1−2α,γ,∞). Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff Hη⁎ measure, where η⁎>0 is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations.

Formation and local stability of a two-dimensional Prandtl boundary layer system in fluid dynamics

By the linearized method, Oleǐnik-Samokhin had constructed solutions to distinguish between two cases of the behavior of the fluid at the initial stage of its motion past the surface in Oleinǐk and Samokhin (1999). In this paper, we consider the development of the boundary layer about a body that gradually starts to move in a resting fluid. In the analytical frame, the formation of the layer shows that, when the layer starts to move, the local solutions of the Prandtl boundary layer system is positive. By the Crocco transformation, the Prandtl boundary layer system reduces a degenerate parabolic equation with a nonlinear boundary value condition. Then by the reciprocal transformation, we can obtain a divergence type parabolic equation. We quote a new kind of BV entropy solution matching up with this divergence type parabolic equation when t≥t0, where t0 is a small enough positive constant. By imposing some close connections between the velocity of the outflow and the geometric characteristic of the spatial domain, using Kružkov’s bi-variables method, the local stability of BV entropy solutions is proved. Thus, by the Crocco inverse transformation, in the analytical frame, we obtain the existence and the uniqueness of the global solution of the Prandtl boundary layer system for a regular spatial domain.

Thermodynamic solution of the homogeneity, isotropy and flatness puzzles (and a clue to the cosmological constant)

We obtain the analytic solution of the Friedmann equation for fully realistic cosmologies including radiation, non-relativistic matter, a cosmological constant λ and arbitrary spatial curvature κ. The general solution for the scale factor a(τ), with τ the conformal time, is an elliptic function, meromorphic and doubly periodic in the complex τ-plane, with one period along the real τ-axis, and the other along the imaginary τ-axis. The periodicity in imaginary time allows us to compute the Hawking temperature and the global gravitational entropy of cosmological spacetimes, just as Gibbons, Hawking and Perry did for black holes and de Sitter spacetime. The global gravitational entropy favours universes like our own which are spatially flat, homogeneous, and isotropic, with a small positive cosmological constant.

Distributed Dynamic Event-Triggered Consensus Control for Multiagent Systems With Guaranteed Performance and Positive Inter-Event Times

This paper addresses the distributed consensus issue of linear multiagent systems (MASs) subject to disturbances. First, a fully distributed adaptive control law is established for each agent, removing the assumption that global information should be known prior. Additionally, unlike existing consensus studies, the restrictions on continual communication among agents and constant controller updates are eliminated through the designed state observers and event-triggered conditions (ETCs). Moreover, an innovative event-triggered condition (ETC) is created that, via the inclusion of a dynamic parameter, captures mixed triggering and time-varying thresholds as special cases. Particularly, distinct from some dwell time methods that enforce inter-event time (IET) greater than zero at the expense of asymptotic stability, positive inter-event times (IETs) and asymptotic stability can be guaranteed simultaneously by the suggested event-triggered conditions (ETCs) in the absence of disturbances. Furthermore, positive minimum inter-event times (MIETs) can be maintained even in the presence of disturbances, which indicates the robustness of the proposed event-triggered mechanism (ETM). It is shown that the suggested technique performs disturbances suppression for all agents in a completely distributed manner while omitting Zeno behavior. Finally, a simulation example demonstrates the suggested scheme’s efficacy. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The multiagent consensus problem is considered in this paper, which can be applied to some practical systems, e.g., the unmanned aerial vehicles formation problems, as the control objective in these applications can be converted into the consensus problem of MASs. Different from existing results, a novel ETM is proposed, reducing agents’ communication and preventing continuous controller updates. In addition, disturbances widely exist in engineering. The proposed method ensures that the interval between two consecutive event triggers is greater than a small positive constant. Therefore, it is helpful for applying the proposed method in practical engineering. Finally, the established method is applied to a two-mass-spring control problem, and the effectiveness is illustrated. It is anticipated that the proposed approach can be extended to MASs with more realistic network-induced constraints and applied to more practical engineering systems.

A single-stage optimization procedure for data envelopment analysis

In data envelopment analysis, a conventional procedure for testing efficiency of decision making units (DMUs) consists of two stages, each requiring solution of a linear program. The first stage identifies the input or output radial efficiency of a DMU, and the second stage maximizes the sum of component input and output slacks. A traditional alternative is the single-stage approximation of the two-stage procedure in which the objective functions of the first and second stages are combined, with the latter multiplied by a small positive constant epsilon. A known drawback of such approach is that very small values of epsilon may cause computational inaccuracies and large values do not allow a good approximation. In this paper, we develop a new single-stage linear programming approach that does not require the specification of a small constant epsilon and is completely equivalent to the conventional two-stage solution procedure. It produces exactly the same sets of primal and dual optimal solutions as the two separate stages of the two-stage approach. The new single-stage procedure is applicable to any convex polyhedral technology.

On fractional sums of the divisor functions

In this paper, we consider the fractional sum of the divisor functions. We can improve previous results considered by [O. Bordellés, On certain sums of number theory, Int. J. Number Theory 18(9) (2022) 2053–2074; K. Liu, J. Wu and Z. S. Yang, On some sums involving the integral part function, preprint (2021); arXiv:2109.01382v1 [math.NT]]. Precisely, we can show that [Formula: see text] where [Formula: see text] is an arbitrary small positive constant and [Formula: see text] is the number of representations of [Formula: see text] as product of [Formula: see text] natural numbers.

On the blow-up criterion for the Navier–Stokes equations with critical time order

We prove that every smooth solution u(t,x) on (0,T) of incompressible Navier–Stokes equations on Rn is extensible beyond t>T if u(t,x)∈Lwr(0,T;Lσp) for 2r+np=1 and p>n satisfies blow-up critical time order estimate: ‖u(t)‖Lp≤ϵ(T−t)−p−n2p for T−δ<t<T with sufficiently small positive constants ϵ and δ. Here Lwr denote the weak Lr space. It is well-known that if the solution u satisfies u∈Lr(0,T;Lσp) with n<p≤∞ and 2≤r<∞ such that 2r+np=1 then u is extensible continued beyond the time T. In this paper, we consider the blow-up criterion when u∉Lr(0,T;Lσp).

Forecast analysis on interacting dark energy models from future generation PICO and DESI missions

ABSTRACT The next-generation cosmic microwave background (CMB) satellite missions are expected to provide robust constraints on a wide range of cosmological parameters with unprecedented precision. But these constraints on the parameters could weaken if we do not attribute dark energy to a cosmological constant. The cosmological models involving interaction between dark energy and dark matter can give rise to comparable energy densities at the present epoch, thereby alleviating the so-called cosmic coincidence problem. In the present paper, we perform a forecast analysis to test the ability of the future generation high-sensitive CMB, and baryon acoustic oscillation (BAO) experiments to constrain phenomenological interacting dark energy models. We consider cosmic variance limited future CMB experiment Probe of Inflation and Cosmic Origins (PICO) along with BAO information from the Dark Energy Spectroscopic Instrument (DESI), to constrain the parameters of the interacting dark sector. Based on the stability of the cosmological perturbations, we consider two possibilities for the interaction scenario. We investigate the impact of both coupling constant and equation of state parameter of dark energy on CMB temperature power spectrum, matter power spectrum, and fσ8. We have used simulated temperature and polarization data from PICO within the multipole ranges (ℓ = 2–4000), and as expected, we do see PICO alone produces better constraints than Planck on the ΛCDM parameters. With the integration of PICO and DESI missions, we observe a significant improvement in the constraints on several cosmological parameters, especially the equation of state parameter of dark energy. However, we note that additional data are required to constrain a small positive coupling constant.

EECDN: Energy-efficient Cooperative DNN Edge Inference in Wireless Sensor Networks

Multi-access edge computing (MEC) is emerging to improve the quality of experience of mobile devices including internet of things sensors by offloading computing intensive tasks to MEC servers. Existing MEC-enabled cooperative computation offloading works focus on the optimization of total energy consumption but fail to exploit multi-relay diversity and min-max fairness of energy consumption on participated sensors. We explore a typical wireless sensor network with multi-source, multi-relay, and one edge server, where relay nodes can provide both cooperative communication and computation services. We divide the energy efficiency optimization problem into two sub-problems: One is to minimize the weighted average total energy consumption per time slot, and the other is to minimize the maximum weighted energy consumption. For the first sub-problem, we propose an optimal algorithm named as optimal weighted average total energy consumption algorithm (OTCA) based on bipartite matching. For the second sub-problem, greedy algorithm for fairness guarantee (GAF) is proposed with an approximation ratio of (1 + ε), where ε is a small positive constant. Extensive numerical results show that OTCA outperforms the baseline algorithms by 26.7–77.4% on the average total weighted energy consumption while GAF outperforms benchmark algorithms by 30.7–84.4%. NS-3 simulation experiments comply with numerical results.

On regularity of weak solutions for the Navier–Stokes equations in general domains

AbstractLet u be a weak solution of the instationary Navier–Stokes equations in a completely general domain which additionally satisfies the strong energy inequality. Firstly, we prove that u is regular if the kinetic energy is left‐side Hölder continuous with Hölder exponent and with a sufficiently small Hölder seminorm. This result extends the previous ones by several authors [5, 6, 7, 8] in which the domain Ω is additionally supposed to be bounded or have the uniform C2‐boundary . Secondly, we show that if and for all with a sufficiently small positive constant C then u is regular in [0, T). Our proofs use the theory about the existence of local strong solutions and uniqueness arguments in the general domain.

Boundedness in asymmetric oscillations under the non-resonant case

In this paper, we are concerned with the boundedness of all solutions of the asymmetric oscillationx″+ax+−bx−=p(t), where x+=max⁡{x,0}, x−=max⁡{−x,0}, p(t) is a real analytic 2π periodic function, a and b are two different positive constants satisfying ω0:=12(1a+1b)∈R\\Q and the condition|kω0−l|≥c0Ω(|k|),k∈Z\\{0},l∈Z, where Ω is an approximation function and c0 is a small positive constant. In particular, when a=5,b=1,p(t)=cos⁡4t, the boundedness of all solutions will be proved.

On the Orbital Elements of the Two-body Problem with Slowly Decreasing Mass: The Gyldén–Mestchersky Cases

Abstract We study the dynamics of two-body problems with gravitational parameters μ = μ(t) decreasing with the time. In particular we focus our attention on the behavior of the orbital elements in the first two Gyldén–Mestchersky cases: in the first case (GM-I) , where μ 0 is the constant initial value of the parameter and α a small positive constant. In the second case (GM-II) . In the GM-I problem, using a suitable transformation of the radius vector and the physical time, which reduces the original problem to a pure Kepler problem, we are able to obtain explicit analytical expressions of the orbital elements that hold true for all and therefore derive their asymptotic behavior. Moreover using the alternative scaled time these expressions show clearly their periodic and secular terms. For the GM-II problem a similar transformation to the GM-I problem leads to a perturbed Kepler problem with a radial direction perturbation. Although exact solutions in terms of elliptic functions may be obtained, the use of some integrals leads to analytical expressions of orbital elements that allow us to study their behavior. Finally, some comparisons between the time evolution of elements in both problems are also included.

Charged strange star with Krori–Barua potential in f(R, T) gravity admitting Chaplygin equation of state

In present paper a new compact star model in f(R,T) gravity is obtained where R and T denote the Ricci scalar and the trace of energy-momentum tensor $T_{\mu \nu}$ respectively. To develop the model we consider the spherically symmetric space-time along with anisotropic fluid distribution in presence of electric field with $f(R,T) = R + 2\gamma T$ where $\gamma$ is a small positive constant. We have used the Chaplygin equation of state to explore the stellar model. The field equations for $f(R,T)$ gravity have been solved by employing the Krori-Barua ansatz already reported in literature [J. Phys. A, Math. Gen. 8:508, 1975]. The exterior spacetime is described by Reissner-Nordstrom line element for smooth matching at the boundary. It is worthwhile to mention here that the values of all the constants involved with this model have been calculated for the strange stars 4U 1538-52 for different values of {\gamma} with the help of matching conditions. The acceptability of the model is discussed in details both analytically and graphically by studying the physical attributes of matter density, pressures, anisotropy factor, stability etc. We have also obtained the numerical values in tabular form for central density, surface density, central pressure and central adiabatic index for different values of $\gamma$. The solutions of the field equations in Einstein gravity can be regained by simply putting $\gamma = 0$ to our solution. Moreover, the proposed model is shown to be physically admissible and corroborate with experimental observations on strange star candidates such as 4U 1538-52.

Seeded graph matching via large neighborhood statistics

We study a noisy graph isomorphism problem, where the goal is to perfectly recover the vertex correspondence between two edge‐correlated graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. We show that it is possible to achieve the information‐theoretic limit of graph sparsity in time polynomial in the number of vertices n. Moreover, we show the number of seeds needed for perfect recovery in polynomial‐time can be as low as in the sparse graph regime (with the average degree smaller than ) and in the dense graph regime, for a small positive constant . Unlike previous work on graph matching, which used small neighborhoods or small subgraphs with a logarithmic number of vertices in order to match vertices, our algorithms match vertices if their large neighborhoods have a significant overlap in the number of seeds.

Existence and stability of bistable wavefronts in a nonlocal delayed reaction–diffusion epidemic system

In this paper, we consider the monotone travelling wave solutions of a reaction–diffusion epidemic system with nonlocal delays. We obtain the existence of monotone travelling wave solutions by applying abstract existence results. By transforming the nonlocal delayed system to a non-delayed system and choosing suitable small positive constants to define a pair of new upper and lower solutions, we use the contraction technique to prove the asymptotic stability (up to translation) of monotone travelling waves. Furthermore, the uniqueness and Lyapunov stability of monotone travelling wave solutions will be established with the help of the upper and lower solution method and the exponential asymptotic stability.

Plasmonic nanoparticles for environmental analysis

Gold and silver nanoparticles have unique optical properties. For instance, the intense colour of nanoparticle suspensions results from the excitation of a collective oscillation of surface conduction electrons, named surface plasmons. This excitation is done using an electromagnetic radiation that interacts with nanoparticles having a negative real and small positive imaginary dielectric constant, such as nanoparticles of gold or silver. The plasmonic optical properties of metal nanostructures are dependent on their shape and size, the dielectric properties of the metal and the surroundings and on the possible electromagnetic coupling with the localized surface plasmons in nearby other plasmonic objects. The other important consequence of the excitation of surface plasmons is a local significant enhancement of the electromagnetic field at some places of the illuminated nanoparticles. Specific plasmonic properties of gold and silver nanoparticles have allowed the development of many sensors for chemical analysis, including sensors dedicated for environmental analysis. Some of these sensors are so sensitive that recording of the reliable analytical signal of a single molecule is possible. Here, we review analytical techniques based on plasmonic properties of metallic nanoparticles for environmental analysis. We present the theory and mechanism of interaction of the electromagnetic radiation with the plasmonic nanoparticles. We detail analytical techniques including methods utilizing local enhancement of the intensity of the electromagnetic field induced by plasmons, and hence increase in the efficiency of some optical processes in the proximity of the plasmonic nanoparticles. Those techniques are surface-enhanced Raman scattering, surface-enhanced infrared absorption and metal-enhanced fluorescence and methods based on the change in the optical properties of plasmonic nanoparticles caused by the analyte-induced aggregation or by analyte-influenced growth or etching of plasmonic nanostructures. Environmental compounds include heavy metal cations, metallo-organic compounds, polycyclic aromatic hydrocarbons, pesticides, nitrite ions, bacterial cells and bacterial pathogens.

Catalytic creation of baby bubble universe with small positive cosmological constant

We investigate the decay of metastable de Sitter, Minkowski and anti-de Sitter vacua catalyzed by a black hole and a cloud of strings. We apply the method to the creation of the four dimensional bubble universe in the five dimensional anti-de Sitter spacetime recently proposed by Banerjee, Danielsson, Dibitetto, Giri and Schillo [1, 2]. We study the bounce action for the creation and find that the bubble with very small cosmological constant, of order Λ(4)/ {M}_4^4 ∼ 10−120, is favored by the catalysis by assuming appropriate mass scales of the black hole and the cloud of strings to reproduce the present energy densities of matter and radiation in the bubble universe.

Relaxation of the cosmological constant

We present a model that naturally tunes a large positive cosmological constant to a small cosmological constant. A slowly rolling scalar field decreases the cosmological constant to a small negative value, causing the universe to contract, thus reheating it. An expanding universe with a small positive cosmological constant can be obtained, respectively, by coupling this solution to any model of a cosmological bounce and coupling the scalar field to a sector that undergoes a technically natural phase transition at the meV scale. A robust prediction of this model is a rolling scalar field today with some coupling to the standard model. This can potentially be experimentally probed in a variety of cosmological and terrestrial experiments, such as probes of the equation of state of dark energy, birefringence in the cosmic microwave background and terrestrial tests of Lorentz violation.

Difference Schemes on Uniform Grids for an Initial-Boundary Value Problem for a Singularly Perturbed Parabolic Convection-Diffusion Equation

Abstract The convergence of difference schemes on uniform grids for an initial-boundary value problem for a singularly perturbed parabolic convection-diffusion equation is studied; the highest x-derivative in the equation is multiplied by a perturbation parameter ε taking arbitrary values in the interval ( 0 , 1 ] {(0,1]} . For small ε, the problem involves a boundary layer of width 𝒪 ⁢ ( ε ) {\mathcal{O}(\varepsilon)} , where the solution changes by a finite value, while its derivative grows unboundedly as ε tends to zero. We construct a standard difference scheme on uniform meshes based on the classical monotone grid approximation (upwind approximation of the first-order derivatives). Using a priori estimates, we show that such a scheme converges as { ε ⁢ N } , N 0 → ∞ {\{\varepsilon N\},N_{0}\to\infty} in the maximum norm with first-order accuracy in { ε ⁢ N } {\{\varepsilon N\}} and N 0 {N_{0}} ; as N , N 0 → ∞ {N,N_{0}\to\infty} , the convergence is conditional with respect to N, where N + 1 {N+1} and N 0 + 1 {N_{0}+1} are the numbers of mesh points in x and t, respectively. We develop an improved difference scheme on uniform meshes using the grid approximation of the first x-derivative in the convective term by the central difference operator under the condition h ≤ m ⁢ ε {h\leq m\varepsilon} , which ensures the monotonicity of the scheme; here m is some rather small positive constant. It is proved that this scheme converges in the maximum norm at a rate of 𝒪 ⁢ ( ε - 2 ⁢ N - 2 + N 0 - 1 ) {\mathcal{O}(\varepsilon^{-2}N^{-2}+N^{-1}_{0})} . We compare the convergence rate of the developed scheme with the known Samarskii scheme for a regular problem. It is found that the improved scheme (for ε = 1 {\varepsilon=1} ), as well as the Samarskii scheme, converges in the maximum norm with second-order accuracy in x and first-order accuracy in t.

Hierarchy Theorems for Testing Properties in Size-Oblivious Query Complexity

Focusing on property testing tasks that have query complexity that is independent of the size of the tested object (i.e., depends on the proximity parameter only), we prove the existence of a rich hierarchy of the corresponding complexity classes. That is, for essentially any function $$q : (0, 1] \rightarrow \mathbb{N}$$ , we prove the existence of properties for which $$\epsilon$$ -testing has query complexity $$\Theta(q(\Theta(\epsilon)))$$ . Such results are proved in three standard domains that are often considered in property testing: generic functions, adjacency predicates describing (dense) graphs, and incidence functions describing bounded-degree graphs. These results complement hierarchy theorems of Goldreich, Krivelevich, Newman, and Rozenberg (Computational Complexity, 2012), which refer to the dependence of the query complexity on the size of the tested object, and focus on the case that the proximity parameter is set to some small positive constant. We actually combine both flavors and get tight results on the query complexity of testing when allowing the query complexity to depend on both the size of the object and the proximity parameter.