Manifold Research Articles

Numerical Study of Large Footprint (24 × 24 mm2) Silicon-Based Embedded Microchannel Three-Dimensional Manifold Coolers

Abstract Silicon-based embedded microchannel with three-dimensional (3D) manifold (MF) μ-cooler offers lower pressure drop and increased heat removal capability (>1 kW/cm2) for microprocessors and power electronics cooling using single-phase water. In this paper, we present a thermal–fluidic numerical analysis of silicon-embedded microchannel cooling. We develop a full-scale computational fluid dynamics (CFD) model of a large footprint (24 × 24 mm2) device having embedded microchannels and a 3D manifold. It is found that the pressure/velocity distributions at three different critical regions inside the inlet manifold have a significant impact on the temperature distribution. A previous study reported a shift of the chip temperature hot-spot at high flow rates; this study delves deep into the flow and pressure variations within the MF and cold plate (CP) that leads to this shift. This study also investigates the degree of flow maldistribution, first between the manifold channels caused by the plenum and then between the cold plate channels caused by individual MF channels. Finally, this study concludes with a comparison between two different 3D manifold inlet channel heights. The comparison reveals that the manifold with 1.5 mm thickness can reduce the pressure drop by a factor of 4 while maintaining the same thermal resistance of 0.04 K cm2/W, thus indicating an increase in the coefficient of performance (COP) by a factor of 4, compared with a manifold thickness of 0.7 mm.

Gauged N=3 , D=4 supergravity: A new web of marginally connected vacua

We analyze the vacuum structure of $\mathcal{N}=3,D=4$ supergravity coupled to 9 vector multiplets with gauge group ${\rm SO}(3)\times {\rm SU}(3)$. Aside from the central $\mathcal{N}=3$ AdS$_4$ vacuum at the origin, on which the supermultiplet structure reproduces the massless sector of M-theory compactified on $\mathrm{N^{0,1,0}}$, we find a rich structure of AdS$_4$ vacua preserving $\mathcal{N}=0,1,2,3$ supersymmetry. These new vacua are arranged in a manifold spanned by scalar fields corresponding to exactly marginal deformations of the dual CFT. This manifold has the form $T^3/K$, where $K$ is a discrete subgroup of the gauge group: $\mathcal{N}=3,2$ and $1$ vacua correspond, respectively, to a point, a line and a surface in the three-dimensional vacuum manifold. We study RG flows from the central $\mathcal{N}=3$ vacuum and elaborate on the possible higher dimensional origin of the new vacua. For the reader's convenience we also provide a review of the embedding tensor formulation of $D=4$, $\mathcal{N}=3$ gauged supergravities. In particular we provide formulas involving the fermion shift tensors and mass matrices in $\mathcal{N}=3$ theories, which can be applied to a generic gauging.

Geometry of positive scalar curvature on complete manifold

Abstract In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. On three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature and width. On higher-dimensional manifold, we obtain a volume growth with a stronger condition.

Three Dimensional Homogeneous Hyperbolic Ricci Solitons

In this paper, we consider the self-similar solutions to the hyperbolic geometric flow, called hyperbolic Ricci solitons. Also, we investigate hyperbolic Ricci solitons on three-dimensional homogeneous manifolds with Riemannian and Lorentzian metrics and obtain some hyperbolic Ricci solitons on such manifolds.

Asymptotic structure with a positive cosmological constant

This is the second of two papers (Fernández-Álvarez F and Senovilla J M M 2021 Class. Quantum Grav. 39 165011) that study the asymptotic structure of space–times with a non-negative cosmological constant Λ. This paper deals with the case Λ > 0. Our approach is founded on the ‘tidal energies’ built with the Weyl curvature and, specifically, we use the asymptotic super-Poynting vector computed from the rescaled Bel–Robinson tensor at infinity to provide a covariant, gauge-invariant, criterion for the existence, or absence, of gravitational radiation at infinity. The fundamental idea we put forward is that the physical asymptotic properties are encoded in , where the first element of the triplet is a three-dimensional manifold, the second is a representative of a conformal class of Riemannian metrics on , and the third element is a traceless symmetric tensor field on . The full set of physically relevant properties of the space–time cannot be characterised at infinity without taking D ab into consideration, and our radiation criterion takes this fully into account. We similarly propose a no-incoming radiation criterion based also on the triplet and on radiant supermomenta deduced from the rescaled Bel–Robinson tensor too. We search for news tensors encoding the two degrees of freedom of gravitational radiation and argue that any news-like object must be associated to, and depends on, two-dimensional cross-sections of . We identify one component of news for every such cross-section and present a general strategy to find the second component, which depends on the particular physical situation. We put in connection the radiation condition and the news-like tensors with the directional structure of the gravitational field at infinity and the criterion of no-incoming radiation. We also introduce the concept of equipped by endowing the conformal boundary with a selected congruence of curves which may be determined by the algebraic structure of the asymptotic Weyl tensor. We also define a group of asymptotic symmetries preserving the new structures. We consider the limit Λ → 0, and apply all our results to selected exact solutions of Einstein field equations in order to illustrate their validity.

Two fast/slow decompositions as well as period-adding sequences in a generalized Bonhoeffer-van der Pol electronic circuit

Bursting dynamics in a generalized Bonhoeffer-van der Pol electronic circuit can reflect multi-timescale characteristics. We use two fast-slow decompositions for the study of the mixed mode oscillations and period-adding sequences observed in this GBVP system. Assuming time-scale ratio clarified by parameter ε, with one fast and two slow variables, a singular periodic orbit can be constructed by solving the desingularized system for the flow on the upper and lower sheets of three-dimensional critical manifold. Such dynamics due to the folded singularity and S-shaped surface. The variation of the parameter values lead to more spikes of the bursting as the upper attracting and center repelling sheets intersect and twist near the folded node. On the other hand, if we choose scale ratio as the other parameter c, two-fast/one-slow analysis could be used. Here, one treats the slow variable as a parameter of the fast subsystem, and studies the bifurcation structure of this subsystem including Bogdanov-Takens (BT), generalized Hopf (GH) and cusp bifurcations. Characteristic features of this case are subcritical Hopf bifurcation and slow passage in close distance to the jump. Furthermore, the cascades of period-adding sequences, resulting in chaotic bursting are observed and discussed in detail.

Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry

Abstract We prove that if an η \eta -Einstein para-Kenmotsu manifold admits a conformal η \eta -Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a conformal η \eta -Ricci soliton is Einstein if its potential vector field V V is infinitesimal paracontact transformation or collinear with the Reeb vector field. Furthermore, we prove that if a para-Kenmotsu manifold admits a gradient conformal η \eta -Ricci almost soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits conformal η \eta -Ricci soliton and satisfy our results. We also have studied conformal η \eta -Ricci soliton in three-dimensional para-cosymplectic manifolds.

On Embedding of the Morse–Smale Diffeomorphisms in a Topological Flow

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse–Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse–Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse–Smale diffeomorphisms to be embedded in topological flows. The progress achieved in solving of Palis’s problem in dimension three is associated with the recently obtained complete topological classification of Morse–Smale diffeomorphisms on threedimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

Nonautonomous vector fields on $$S^3$$: Simple dynamics and wild embedding of separatrices

We construct new substantive examples of non-autonomous vector fields on 3-dimensional sphere having a simple dynamics but non-trivial topology. The construction is based on two ideas: the theory of diffeomorpisms with wild separatrix embedding (Pixton, Bonatti-Grines, etc.) and the construction of a non-autonomous suspension over a diffeomorpism (Lerman-Vainshtein). As a result, we get periodic, almost periodic or even nonrecurrent vector fields which have a finite number of special integral curves possessing exponential dichotomy on $\R$ such that among them there is one saddle integral curve (with an exponential dichotomy of the type (3,2)) having wildly embedded two-dimensional unstable separatrix and wildly embedded three-dimensional stable manifold. All other integral curves tend, as $t\to \pm \infty,$ to these special integral curves. Also we construct another vector fields having $k\ge 2$ special saddle integral curves with tamely embedded two-dimensional unstable separatrices forming mildly wild frames in the sense of Debrunner-Fox. In the case of periodic vector fields, corresponding specific integral curves are periodic with the period of the vector field, and they are almost periodic for the case of almost periodic vector fields.

Knotted 4-regular graphs: Polynomial invariants and the Pachner moves

In loop quantum gravity, states of quantum geometry are represented by classes of knotted graphs, equivalent under diffeomorphisms. Thus, it is worthwhile to enumerate and distinguish these classes. This paper looks at the case of 4-regular graphs, which have an interpretation as objects dual to triangulations of three-dimensional manifolds. Two different polynomial invariants are developed to characterize these graphs—one inspired by the Kauffman bracket relations and the other based on quandles. How the latter invariant changes under the Pachner moves acting on the graphs is then studied.

Intake Manifold Flow Study using Numerical Method with all Runners Opened

Intake manifolds have to be designed to improve engine performance by avoiding the phenomena like inter-cylinder robbery of charge, inertia of the flow in the individual branch pipes, resonance of the air masses in the pipes and the Helmholtz effect. The objective of work is to predict and analyze the flow through intake manifold of four cylinder spark ignition engine. One of the important factors is air flow inside the intake manifold; the ideal intake manifold distributes flow evenly to the piston valves. The structural analysis has been conducted to decide the thickness and material that is suitable for intake manifold to withstand bursting pressure. Three-dimensional inlet manifold was modeled in ANSYS workbench and numerically analyzed by using the commercially available FLUENT software to study the pressure, velocity and flow characteristics inside the runner. The steady state analysis has been carried out for three for All runners open, The predicted results of total pressure loss and total outlet mass flow were discussed. Inlet pipe and plenum connection creates a back step geometry which causes more total pressure loss due to flow recirculation in conventional model. Tapering the geometry is causing more inlet mass flow due to reduction in total pressure loss in the plenum chamber.

Inflationary Universe from Anomaly-FreeFR-Gravity

By adding a three-dimensional manifold to an eleven-dimensional manifold in supergravity, we obtain the action ofFR-gravity and find that it is anomaly-free. We calculate the scale factor of the inflationary universe in this model and observe that it is related to the slow-roll parameters. The tensor-scalar ratioRtensor−scalaris in good agreement with experimental data.

Thermodynamic geometry of spin-one lattice models. I. Spin and quadrupolar orders and critical scaling functions in one dimension.

State space Riemannian geometry is obtained for the one-dimensional Blume-Emery-Griffiths model and its Blume-Capel and Griffiths model limits, and its (pseudo)critical as well as noncritical parameter regimes are extensively investigated. Two codimension one geometries are obtained by taking suitable hypersurfaces in the three-dimensional state space manifold, and the induced thermal metrics are accordingly interpreted in terms of constrained fluctuations. The three-dimensional scalar curvature and the two two-dimensional curvatures are shown to be consistent with Ruppeiner's conjecture relating the inverse of the singular free energy to the thermodynamic scalar curvature. Moreover, they are found to be in an excellent agreement over a greater part of the noncritical region with the corresponding correlation lengths for the spin and the quadrupolar order parameters. The scaling function for the free energy near the pseudocritical and tricritical points is obtained thermodynamically by using Ruppeiner's conjecture. A connection is made between the sign change in the curvatures and the change in fluctuation patterns of the order parameters. In the accompanying paper we shall analyze the geometry of the spin-one model in its mean field approximation.

Consistent truncations around half-maximal AdS5 vacua of 11-dimensional supergravity

We use exceptional field theory to systematically study all possible consistent truncations around any M-theory half-maximal vacua of the warped product form AdS5 × M 3 × S 2 × S 1, with M 3 a three-dimensional manifold. We show that, for any of these vacua, only truncations with at most three vector multiplets are consistent. Furthermore, the possible gaugings are restricted to be either SU(2) × U(1) or ISO(3) × U(1) where, in the first case, the U(1) factor can have different embeddings inside the global symmetry group SO(5, n), where n ⩽ 3 equals the number of vector multiplets. This rules out the possibility of any other gauging arising as a consistent truncation around the aforementioned M-theory vacua. Our analysis shows that of the many flows from half-maximal to quarter-maximal AdS5 vacua constructed in five-dimensional supergravity in Bobev (2018 J. High Energy Phys. 6 86), only those corresponding to an adjoint mass deformation in the dual SCFT can be uplifted to 11-dimensional supergravity. The other flows are five-dimensional artefacts without a higher-dimensional origin. Furthermore, consistent truncations with vector multiplets exist only if the vacuum satisfies certain conditions, which we derive.

∗-Ricci Tensor onα-Cosymplectic Manifolds

In this paper, we study α-cosymplectic manifoldMadmitting∗-Ricci tensor. First, it is shown that a∗-Ricci semisymmetric manifoldMis∗-Ricci flat and aϕ-conformally flat manifoldMis anη-Einstein manifold. Furthermore, the∗-Weyl curvature tensorW∗onMhas been considered. Particularly, we show that a manifoldMwith vanishing∗-Weyl curvature tensor is a weakϕ-Einstein and a manifoldMfulfilling the conditionRE1,E2⋅W∗=0isη-Einstein manifold. Finally, we give a characterization for α-cosymplectic manifoldMadmitting∗-Ricci soliton given as to be nearly quasi-Einstein. Also, some consequences for three-dimensional cosymplectic manifolds admitting∗-Ricci soliton and almost∗-Ricci soliton are drawn.

Free boundary constant mean curvature surfaces in a strictly convex three-manifold

Let C be a strictly convex domain in a three-dimensional Riemannian manifold with sectional curvature bounded above by a constant, and let $$\Sigma $$ be a constant mean curvature surface with free boundary in C. We provide a pinching condition on the length of the traceless second fundamental form on $$\Sigma $$ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if C is a geodesic ball of three-dimensional space forms, then $$\Sigma $$ is either a spherical cap or a Delaunay surface.

The Schouten-van Kampen connection on quasi-Sasakian manifolds

"In the present paper, we study three-dimensional quasi-Sasakian manifolds admitting the Schouten-van Kampen connection. We characterize quasi-Sasakian manifolds and nd certain curvature properties with respect to the Schouten-van Kampen connection. Finally, we construct an example of a three-dimensional quasi-Sasakian manifold admitting the Schouten-van Kampen connection which veri es the results discussed in the present paper."

On the connectivity of spaces of three-dimensional domino tilings

We consider domino tilings of three-dimensional cubiculated manifolds with or without boundary, including subsets of Euclidean space and three-dimensional tori. In particular, we are interested in the connected components of the space of tilings of such regions under local moves. Building on the work of the third and fourth authors we allow two possible local moves, the flip and trit. These moves are considered with respect to two topological invariants, the twist and flux. Our main result proves that, up to refinement, ∙ \bullet \; Two tilings are connected by flips and trits if and only if they have the same flux. ∙ \bullet \; Two tilings are connected by flips alone if and only if they have the same flux and twist.

$$L^p$$ Regularity Estimates for a Class of Integral Operators with Fold Blowdown Singularities

We prove sharp \(L^p\) regularity results for a class of generalized Radon transforms for families of curves in a three-dimensional manifold associated with a canonical relation with fold and blowdown singularities. The proof relies on decoupling inequalities by Wolff and Bourgain–Demeter for plate decompositions of thin neighborhoods of cones and \(L^2\) estimates for related oscillatory integrals.

Building three-dimensional differentiable manifolds numerically

A method is developed here for building differentiable three-dimensional manifolds on multicube structures. This method constructs a sequence of reference metrics that determine differentiable structures on the cubic regions that serve as non-overlapping coordinate charts on these manifolds. It uses solutions to the two- and three-dimensional biharmonic equations in a sequence of steps that increase the differentiability of the reference metrics across the interfaces between cubic regions. This method is algorithmic and has been implemented in a computer code that automatically generates these reference metrics. Examples of three-manifolds constructed in this way are presented here, including representatives from five of the eight Thurston geometrization classes, plus the well-known Hantzsche-Wendt, the Poincaré dodecahedral space, and the Seifert-Weber space.