Numerical and experimental studies on the viscous folding in diverging microchannels

We performed numerical and experimental studies on the viscous folding in diverging microchannel flows which were recently reported by Cubaud and Mason (Phys Rev Lett 96:114501, 2006a). We categorized the flow patterns as “stable”, “folding,” and “chaotic” depending on channel shape, flow ratio, and viscosity ratio between two fluids. We focused on the effect of kinematic history on viscous folding, in particular, by changing the shape of diverging channels: 90°, 45°, and hyperbolic channel. In experiments, the proposed power–law relation (f ~ [(g)\dot]¹, f\sim \dot{\gamma }^{1}, where f is the folding frequency, and [(g)\dot] \dot{\gamma } is the characteristic shear rate) by Cubaud and Mason (Phys Rev Lett 96:114501, 2006a) was found to be valid even for hyperbolic channel. The hyperbolic channel generated moderate flows with smaller folding frequency, amplitude, and a delay of onset of the folding compared with other two cases, which is considered to be affected by compressive stress when compared to the simulation results. In each channel, the folding frequency increases and the amplitude decreases as the thread width decreases since higher compressive stress is applied along the thin thread. The secondary folding was also reproduced in the simulation, which was attributed to locally heterogeneous development of compressive stresses along the thread. This study proves that the viscous folding can be controlled by the design of flow kinematics and of the compressive stresses at the diverging region.     

Soft set theory applied to p-ideals of BCI-algebras related to fuzzy points

The notions of $(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})The notions of ([`( ? )],[`( ? )] ú[`q])(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})-fuzzy p-ideals and fuzzy p-ideals with thresholds related to soft set theory are discussed. Relations between ([`( ? )],[`( ? )] ú[`q])(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})-fuzzy ideals and ([`( ? )],[`( ? )] ú[`q])(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})-fuzzy p-ideals are investigated. Characterizations of an ([`( ? )],[`( ? )] ú[`q])(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})-fuzzy p-ideal and a fuzzy p-ideal with thresholds are displayed. Implication-based fuzzy p-ideals are discussed.     

Arthur and Merlin as Oracles

We study some problems solvable in deterministic polynomial time given oracle access to the (promise version of) Arthur–Merlin class. Our main results are the following:

Erlangen pipe flow: the concept and DNS results for microflow control of near-wall turbulence

The concept of a micropatterned surface morphology capable of producing self-stabilization of turbulence in wall-bounded flows is considered in pipes of non-circular cross-sections which act to restructure fluctuations towards the limiting state where these must be entirely suppressed. Direct numerical simulations of turbulence in pipes of polygon-shaped cross-sections with straight and profiled sides were performed at a Reynolds number $Re_\tau \simeq {\mathrm 300}$ based on the wall shear velocity and the hydraulic diameter. Using the lattice Boltzmann numerical algorithm, turbulence was resolved with up to ${\mathrm 540\times 10^6}$ grid points ( ${\mathrm 8,192\times 257 \times 256}$ in the x ₁, x ₂ and x ₃ directions). The DNS results show a decrease in the viscous drag around corners, resulting in a reduction of the skin-friction coefficient compared with expectations based on the well-established concept of hydraulic diameter and the use of the Blasius correlation. These findings support the conjecture that turbulence might be completely suppressed if the pipe cross-section is a polygon consisting of a sufficient number of profiled sides of the same length which intersect at right angles at the corners.     

Generalized fuzzy interior ideals of semigroups

The concept of $(\overline{\in},\overline{\in} \vee \overline{q})The concept of ([`( ? )],[`( ? )] ú[`(q)])(\overline{\in},\overline{\in} \vee \overline{q})-fuzzy interior ideals of semigroups is introduced and some related properties are investigated. In particular, we describe the relationships among ordinary fuzzy interior ideals, (∈, ∈ ∨ q)-fuzzy interior ideals and ([`( ? )],[`( ? )] ú[`(q)])(\overline{\in},\overline{\in} \vee \overline{q})-fuzzy interior ideals of semigroups. Finally, we give some characterization of [F] _t by means of (∈, ∈ ∨ q)-fuzzy interior ideals.     

The holy grail of microfluidics: sub-laminar drag by layout of periodically embedded microgrooves

The sub-laminar drag effect of microgroove surfaces was studied numerically in a steady two-dimensional channel flow at subcritical Reynolds numbers. Considerations are restricted to grooves of a few viscous length scales in depth, which are assumed not to promote the laminar to turbulent transition process. It was found that the drag reduction effect is due to the layout of grooves with respect to the flow direction and contour geometry. Results of computations show that for grooves of curved contour placed normal to the flow direction, drag arising from viscous and pressure forces is modulated due to the functional dependence of forces on the surface area projected in the flow direction. Such a groove layout leads to a large skin-friction reduction, but a comparable increase in pressure drag results in sub-laminar drag if drag over flat surface is considered as a reference. For a curved groove contour, the drag reduction increases with increasing Reynolds number and reaches about 5 % at Reynolds numbers approaching critical.     

Interpolation of Shifted-Lacunary Polynomials

Given a “black box” function to evaluate an unknown rational polynomial f ? \mathbbQ[x]f \in {\mathbb{Q}}[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity $t \in {\mathbb{Z}}_{>0}$t \in {\mathbb{Z}}_{>0}, the shift a ? \mathbbQ\alpha \in {\mathbb{Q}}, the exponents 0 ￡ e₁ < e₂ < ? < e_t{0 \leq e_{1} < e_{2} < \cdots < e_{t}}, and the coefficients c₁, ?, c_t ? \mathbbQ \{0}c_{1}, \ldots , c_{t} \in {\mathbb{Q}} \setminus \{0\} such that

Distributed algorithms for ultrasparse spanners and linear size skeletons

We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an ${O(2^{{\rm log}^{*} n} {\rm log} n)}We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an O(2^{log* n} log n){O(2^{{\rm log}^{*} n} {\rm log} n)} -spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)-size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)-spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size, namely O(n(loglogn)^f){O(n(\log \log n)^{\phi})} , where f = (1 + ?5)/2{\phi = (1 + \sqrt{5})/2} is the golden ratio. As the distance increases the multiplicative distortion of a Fibonacci spanner passes through four discrete stages, moving from logarithmic to log-logarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion.     

Error Analysis for hp-FEM Semi-Lagrangian Second Order BDF Method for Convection-Dominated Diffusion Problems

We present in this paper an analysis of a semi-Lagrangian second order Backward Difference Formula combined with hp-finite element method to calculate the numerical solution of convection diffusion equations in ℝ². Using mesh dependent norms, we prove that the a priori error estimate has two components: one corresponds to the approximation of the exact solution along the characteristic curves, which is O(Dt²+h^m+1(1+\frac\mathopen|logh|Dt))O(\Delta t^{2}+h^{m+1}(1+\frac{\mathopen{|}\log h|}{\Delta t})); and the second, which is O(Dt^p+|| [(u)\vec]-[(u)\vec]_h||_L^￥)O(\Delta t^{p}+\| \vec{u}-\vec{u}_{h}\|_{L^{\infty}}), represents the error committed in the calculation of the characteristic curves. Here, m is the degree of the polynomials in the finite element space, [(u)\vec]\vec{u} is the velocity vector, [(u)\vec]_h\vec{u}_{h} is the finite element approximation of [(u)\vec]\vec{u} and p denotes the order of the method employed to calculate the characteristics curves. Numerical examples support the validity of our estimates.     

Lower Bounds for Agnostic Learning via Approximate Rank

We prove that the concept class of disjunctions cannot be pointwise approximated by linear combinations of any small set of arbitrary real-valued functions. That is, suppose that there exist functions f₁, ?, f_r\phi_{1}, \ldots , \phi_{r} : {− 1, 1}ⁿ → \mathbbR{\mathbb{R}} with the property that every disjunction f on n variables has $\|f - \sum\nolimits_{i=1}^{r} \alpha_{i}\phi _{i}\|_{\infty}\leq 1/3$\|f - \sum\nolimits_{i=1}^{r} \alpha_{i}\phi _{i}\|_{\infty}\leq 1/3 for some reals a₁, ?, a_r\alpha_{1}, \ldots , \alpha_{r}. We prove that then $r \geq exp \{\Omega(\sqrt{n})\}$r \geq exp \{\Omega(\sqrt{n})\}, which is tight. We prove an incomparable lower bound for the concept class of decision lists. For the concept class of majority functions, we obtain a lower bound of W(2ⁿ/n)\Omega(2^{n}/n) , which almost meets the trivial upper bound of 2ⁿ for any concept class. These lower bounds substantially strengthen and generalize the polynomial approximation lower bounds of Paturi (1992) and show that the regression-based agnostic learning algorithm of Kalai et al. (2005) is optimal.     

Quantum search in a possible three-dimensional complex subspace

Suppose we are given an unsorted database with N items and N is sufficiently large. By using a simpler approximate method, we re-derive the approximate formula cos² Φ, which represents the maximum success probability of Grover’s algorithm corresponding to the case of identical rotation angles f = q{\phi=\theta} for any fixed deflection angle F ? [0,p/2){\Phi \in\left[0,\pi/2\right)}. We further show that for any fixed F ? [0,p/2){\Phi \in\left[0,\pi/2\right)}, the case of identical rotation angles f = q{\phi=\theta} is energetically favorable compared to the case |q- f| >> 0{\left|{\theta - \phi}\right|\gg 0} for enhancing the probability of measuring a unique desired state.     

A System of Relational Syllogistic Incorporating Full Boolean Reasoning

We present a system of relational syllogistic, based on classical propositional logic, having primitives of the following form: $$\begin{array}{ll}\mathbf{Some}\, a \,{\rm are} \,R-{\rm related}\, {\rm to}\, \mathbf{some} \,b;\\ \mathbf{Some}\, a \,{\rm are}\,R-{\rm related}\, {\rm to}\, \mathbf{all}\, b;\\ \mathbf{All}\, a\, {\rm are}\,R-{\rm related}\, {\rm to}\, \mathbf{some}\, b;\\ \mathbf{All}\, a\, {\rm are}\,R-{\rm related}\, {\rm to}\, \mathbf{all} \,b.\end{array}$$ Such primitives formalize sentences from natural language like ‘All students read some textbooks’. Here a, b denote arbitrary sets (of objects), and R denotes an arbitrary binary relation between objects. The language of the logic contains only variables denoting sets, determining the class of set terms, and variables denoting binary relations between objects, determining the class of relational terms. Both classes of terms are closed under the standard Boolean operations. The set of relational terms is also closed under taking the converse of a relation. The results of the paper are the completeness theorem with respect to the intended semantics and the computational complexity of the satisfiability problem.     

Communication Complexity Under Product and Nonproduct Distributions

We solve an open problem in communication complexity posed by Kushilevitz and Nisan (1997). Let R_∈(f) and $D^\mu_\in (f)$D^\mu_\in (f) denote the randomized and μ-distributional communication complexities of f, respectively (∈ a small constant). Yao’s well-known minimax principle states that $R_{\in}(f) = max_\mu \{D^\mu_\in(f)\}$R_{\in}(f) = max_\mu \{D^\mu_\in(f)\}. Kushilevitz and Nisan (1997) ask whether this equality is approximately preserved if the maximum is taken over product distributions only, rather than all distributions μ. We give a strong negative answer to this question. Specifically, we prove the existence of a function f : {0, 1}ⁿ ×{0, 1}ⁿ ? {0, 1}f : \{0, 1\}^n \times \{0, 1\}^n \rightarrow \{0, 1\} for which max_{μ product} {D^m _? (f)} = Q(1)  but R _? (f) = Q(n)\{D^\mu_\in (f)\} = \Theta(1) \,{\textrm but}\, R_{\in} (f) = \Theta(n). We also obtain an exponential separation between the statistical query dimension and signrank, solving a problem previously posed by the author (2007).     

The impact of surface roughness on flow through a rectangular microchannel from the laminar to turbulent regimes

Modifications of fluid flow within microscale flow passages by internal surface roughness is investigated in the laminar, transitional, and turbulent regimes using pressure-drop measurements and instantaneous velocity fields acquired by microscopic particle-image velocimetry (micro-PIV). The microchannel under study is rectangular in cross-section with an aspect ratio of 1:2 (depth: width) and a hydraulic diameter of D_h = 600 \upmu m.D_{\rm h} =600\,\upmu \hbox{m}. Measurements are first performed under smooth-wall conditions to establish the baseline flow characteristics within the microchannel followed by measurements for two different rough-wall cases [with RMS roughness heights of 7.51 \upmu m7.51\,\upmu \hbox{m} (0.0125D _h) and 15.1 \upmu m15.1\,\upmu \hbox{m} (0.025D _h)]. The roughness patterns under consideration are unique in that they are reminiscent of surface irregularities one might encounter in practical microchannels due to imperfect fabrication methods. The pressure-drop results reveal the onset of transition above Re_cr=1,800Re_{\rm cr}=1{,}800 for the smooth-wall case, consistent with the onset of transition at the macroscale, along with deviation from laminar behavior at progressively lower Re with increasing roughness. Mean velocity profiles computed from the micro-PIV ensembles at various Re for each surface condition confirm these trends, meaning Re_crRe_{\rm cr} is a strong function of roughness. The ensembles of velocity fields at each Re and surface condition in the transitional regime are subdivided into fields embodying laminar behavior and fields containing disordered motions. This decomposition reveals a clear hastening of the flow toward a turbulent state due both to the roughness dependence of Re _cr and an enhancement in the growth rate of the non-laminar fraction of the flow when the flow is in the early stages of transition. Nevertheless, the range of Re relative to Re _cr over which the flow transitions from a laminar to a turbulent state is found to be essentially the same for all three surface conditions. From a structural viewpoint, instantaneous velocity fields embodying disordered behavior in the transitional regime are found to contain large-scale motions consistent with hairpin-vortex packets irrespective of surface condition. These observations are in accordance with the characteristics of transitional and turbulent flows at the macroscale and therefore indicate that the overall structural paradigm of the flow is relatively insensitive to roughness. From a quantitative viewpoint, however, the intensity of both the velocity fluctuations and structural activity appear to increase substantially with increasing roughness, particularly in the latter stages of transition. These differences are further supported by the trends of single-point statistics of the non-laminar ensembles and quadrant analysis in which an intensification of the velocity fluctuations by surface roughness is noted in the region close to the wall, particularly for the wall-normal fluctuations.     

Multi-party quantum state sharing of an arbitrary two-qubit state with Bell states

We present a new scheme for sharing an arbitrary two-qubit quantum state with n agents. In our scheme, the sender Alice first shares n Einsein-Podolsky-Rosen (EPR) pairs in Bell states with n agents. After setting up the secure quantum channel, Alice first applies (n − 2) Controlled-Not (CNOT) gate operations, and then performs two Bell-state measurements and (n − 2) single-particle measurements (n >2). In addition, all controllers only hold one particle in their hands, respectively, and thus they only need to perform a single-particle measurement on the respective particle with the basis {|0?, |1?}{\{{\vert}0\rangle, {\vert}1\rangle\}}. Compared with other schemes with Bell states, our scheme needs less qubits as the quantum resources and exchanges less classical information, and thus obtains higher total efficiency.     

The sudden death of entanglement in constructed Yang–Baxter systems

Some Hamiltonians are constructed from the unitary \checkR_i,i+1(q, j){\check{R}_{i,i+1}(\theta, \varphi)}-matrices, where θ and j{\varphi} are time-independent parameters. We show that the entanglement sudden death (ESD) can happen in these closed Yang–Baxter systems. It is found that the ESD is not only sensitive to the initial condition, but also has a great connection with different Yang–Baxter systems. Especially, we find that the meaningful parameter j{\varphi} has a great influence on ESD.     

New regime of droplet generation in a T-shape microfluidic junction

We present an experimental study of a new regime of monodisperse micro-droplet generation that we named the balloon regime. A dispersion of oil in water in a T-junction microfluidic system was studied. Several microfluidic devices having different cross-sections of the continuous and the dispersed phases micro-channels were tested. This new regime appears only for low- dispersed phase velocity. The micro-droplet size is mainly related to the geometry of the T-junction micro-channels especially its width and depth, and independent of the continuous and dispersed phases velocities. In our experiments, the velocities of the continuous and the dispersed phases $\overline v_{\rm c}$ and $\overline v_{\rm d}$ respectively, have been varied in a wide range: $\overline v_{\rm c}$ from 0.5 to 500 mm/s, and $\overline v_{\rm d}$ from 0.01 to 30 mm/s. We show that the continuous phase only controls the micro-droplet density, while the dispersed phase linearly changes the frequency of the micro-droplet generation. Another particularity of the present regime, which differentiates it from all other known regimes, is that the micro-droplet retains its circular shape throughout its formation at the T junction, and undergoes no deformation due to the drag forces. We propose a mechanism to explain the formation of micro-droplets in this new regime.     

Redundant modeling in permutation weighted constraint satisfaction problems

In classical constraint satisfaction, redundant modeling has been shown effective in increasing constraint propagation and reducing search space for many problem instances. In this paper, we investigate, for the first time, how to benefit the same from redundant modeling in weighted constraint satisfaction problems (WCSPs), a common soft constraint framework for modeling optimization and over-constrained problems. Our work focuses on a popular and special class of problems, namely, permutation problems. First, we show how to automatically generate a redundant permutation WCSP model from an existing permutation WCSP using generalized model induction. We then uncover why naively combining mutually redundant permutation WCSPs by posting channeling constraints as hard constraints and relying on the standard node consistency (NC*) and arc consistency (AC*) algorithms would miss pruning opportunities, which are available even in a single model. Based on these observations, we suggest two approaches to handle the combined WCSP models. In our first approach, we propose m\text -NC_{\text c}^*m\text {-NC}_{\text c}^* and m\text -AC_{\text c}^*m\text {-AC}_{\text c}^* and their associated algorithms for effectively enforcing node and arc consistencies in a combined model with m sub-models. The two notions are strictly stronger than NC* and AC* respectively. While the first approach specifically refines NC* and AC* so as to apply to combined models, in our second approach, we propose a parameterized local consistency LB(m,Φ). The consistency can be instantiated with any local consistency Φ for single models and applied to a combined model with m sub-models. We also provide a simple algorithm to enforce LB(m,Φ). With the two suggested approaches, we demonstrate their applicabilities on several permutation problems in the experiments. Prototype implementations of our proposed algorithms confirm that applying 2\text -NC_{\text c}^*,  2\text -AC_{\text c}^*2\text {-NC}_{\text c}^*,\;2\text {-AC}_{\text c}^*, and LB(2,Φ) on combined models allow far more constraint propagation than applying the state-of-the-art AC*, FDAC*, and EDAC* algorithms on single models of hard benchmark problems.     

Estimating temperature fluctuations in the early universe

A lagrangian for a k-essence field is constructed for a constant scalar potential, and its form is determined when the scale factor is very small as compared to the present epoch but very large as compared to the inflationary epoch. This means that one is already in an expanding and flat universe. The form is similar to that of an oscillator with time-dependent frequency. Expansion is naturally built into the theory with the existence of growing classical solutions of the scale factor. The formalism allows one to estimate the temperature fluctuations of the background radiation at these early stages (as compared to the present epoch) of the Universe. If the temperature is T _a at time t _a and T _b at time t _b (t _b > t _a ), then, for small times, the probability evolution for the logarithm of the inverse temperature can be estimated as

On the solution of the fuzzy Sylvester matrix equation

In this paper, we consider the fuzzy Sylvester matrix equation AX+XB=C,AX+XB=C, where A ? \mathbbR^{n ×n}A\in {\mathbb{R}}^{n \times n} and B ? \mathbbR^{m ×m}B\in {\mathbb{R}}^{m \times m} are crisp M-matrices, C is an n×mn\times m fuzzy matrix and X is unknown. We first transform this system to an (mn)×(mn)(mn)\times (mn) fuzzy system of linear equations. Then, we investigate the existence and uniqueness of a fuzzy solution to this system. We use the accelerated over-relaxation method to compute an approximate solution to this system. Some numerical experiments are given to illustrate the theoretical results.