Homotopy Research Articles

Stress state of billet – mandrel system during production of hollow steel billet in a unit of continuous casting and deformation. Part 1

The substantiation of the relevance of obtaining continuous cast steel pipe hollow billet is given from the position of improving the quality of pipes made of carbon and alloy steels is given. The article presents an assessment of the quality of the inner surface of pipes made of solid steel pipe billet. A new technology is proposed for the production of hollow steel pipe billets on a resource-saving unit of combined continuous casting and deformation. The photo of the continuous casting and deformation section of JSC Ural Pipe Plant is given, where a pilot unit of combined continuous casting and deformation is installed. The paper presents the results of a theoretical study of stress-strain state of the mandrel and sections of a pipe billet when it is compressed by the strikers of the unit of combined continuous casting and deformation. The authors discuss the general model of the mold – striker system. The initial data on calculation, dimensions of the hollow pipe billet and a description of the calibration of strikers for compression of a hollow steel billet are given. The temperature field of a hollow billet was determined. To simulate the stress-strain state of the metal in the roll pass and the mandrel, four contact pairs were considered. Calculations were made by the finite element method. The dimensions of the final element in the roll pass of hollow billet were determined. The authors established the values and patterns of changes in metal displacements and axial stresses in the roll pass during the production of hollow steel billets in the unit of combined continuous casting and deformation (strikers are made along a constant radius). The stress state of metal in the roll pass was assessed from the standpoint of improving the quality of hollow steel billets.

Creep Instability Mechanism and Control Technology of Soft Coal Roadways Based on Fracture Evolution Law

To address the challenging issues of large deformation, control difficulties, and susceptibility to failure in the support structure of soft coal roadways, this study utilizes the CVISC block creep model in UDEC software. The model incorporates Coulomb slip without cohesive contact to simulate the characteristics of soft coal, such as its loose, fragile, and small-block nature. Additionally, a soft coal nonlinear discrete element creep model is developed to investigate the creep characteristics of soft coal under triaxial compression, with the aim of revealing the underlying creep destabilization mechanism in soft coal tunnels. Based on the research findings, a primary, strong active support technology is proposed. This approach involves the use of high-preload, high-strength anchor rods and anchor cables, as well as the implementation of steel mesh and plastic woven mesh to enhance surface protection. The study highlights that: (1) The shear cracks inside the coal body of the soft coal specimen transform into tensile cracks under external force, leading to an increase in the number of tensile cracks. This is an important symbol of accelerated creep in soft coal. Improvement in peripheral pressure helps inhibit the generation of tensile cracks inside the specimen. (2) The rapid development of creep and inter-particle tensile fissures within the coal body particles themselves, along with the change in stress state after the excavation of the roadway, are the main reasons for the overall creep damage of the roadway. (3) The support force in the early stage of shed cable support is small, which cannot inhibit the accelerated development of tensile fissures. This leads to continuous deformation of the roadway, resulting in the failure of the support structure in the later stage and aggravated roadway damage. (4) The new support technology helps control surface deformation by enhancing the strength of the roadway protection surface. This suppresses the development speed and number of tensile fissures during roadway deformation, improves the starting strength of the roadway for accelerated creep, and enables effective control of the overall deformation of the soft coal roadway. Thus, the effectiveness of roadway support is remarkable.

A new approach for evaluating continuous and discontinuous pipeline deformation induced by soil tunnel excavation

The deformation of the overlying pipeline caused by the soil tunnel excavation cannot be ignored in the case of the small spacing between the pipeline and the tunnel. Based on the rigid bar method, the pipeline-soil interaction model was established, with the simply supported beam as the basic system, and the loads acting on the pipeline by the soil are considered to be linearly distributed. Calculation methods for continuous and discontinuous pipeline deformations were established. The results calculated by the proposed method agree well with the experimental data of centrifuge tests and field data. Parametric study on the effect of the volume loss (η = 1%, 2%,3 %), rotational stiffness (β0 = 4.47 × 106N⋅m/rad, 4.47 × 108N⋅m/rad, 4.47 × 1010N⋅m/rad), ratio of pipeline section length to inflection point of soil settlement curve (L/is = 0.5, 1.0, 1.5, 2.0) and soil elastic modulus (E = 10 MPa, 30 MPa, 50 MPa) on the deflection and joint rotation angle of the discontinuous pipeline were carried out. Results show that: (1) the maximum pipeline deflection and the maximum rotation angle of the joint increase as η increases and decrease as β0 increases; (2) in the "odd" case, the maximum pipeline deflection and the maximum rotation angle of the joint first increase and then decrease as L/is increases, reaching a peak at L/is = 1.5, while in the "even" case, the maximum pipeline deflection decreases as L/is increases and the maximum rotation angle of the joint first increases and then decreases as L/is increases; (3) in the "odd" case, the maximum pipeline deflection and the maximum angle of rotation of the joint decrease as E increases, while the opposite trend is observed in the "even" case. Additionally, the maximum pipeline deflection and the maximum rotation angle of the joint are always greater in the "odd" case than that in the "even" case.

Assessing the predictive capabilities of precipitation strengthening models for deformation twinning in Mg alloys using phase-field simulations

Precipitation strengthening is a key strategy for improving the overall mechanical properties of Mg alloys. In Mg-Al alloys, basal precipitates are known to strengthen against twinning, resulting in an increase in the critical resolved shear stress (CRSS) necessary for continued deformation. Although several models have been proposed to quantify the influence of precipitate shape, size, and distribution on the CRSS, the accuracy, scope, and applicability of these models has not been fully assessed. Accordingly, the objectives of this study are: (i) to analyze the accuracy of analytical models proposed in the literature for precipitation strengthening against twin thickening and propagation (in Mg-Al alloys) using phase-field (PF) simulations, (ii) to propose modifications to these model forms to better capture the observed trends in the PF data, and (iii) to subsequently test the predictiveness of the extended models in extrapolating to experimental strengthening data. First, using an atomistically-informed phase-field method, the interactions between migrating twin boundaries (during the propagation and thickening stages) and basal plates are simulated for different precipitate sizes and arrangements. In general, comparison of the increase in CRSS determined from the PF simulations and the predictions from four precipitation strengthening models reveals that modifications are necessary to the model forms to extend their applicability to precipitation strengthening against both twin thickening and propagation. A subsequent comparison between predictions from the extended models and experimental strengthening data for peak age-hardened samples reveals that the (extended) single dislocation and dislocation wall models provide reasonably accurate values of the increase in CRSS. Ultimately, the results presented here help elucidate the fidelity and applicability of the various hardening models in predicting precipitation strenghtening effects in technologically important alloys.

Generation of Homotopy Classes for Unconstrained 3D Wire Routing from Characteristic Loops

Routing of wire harnesses in complex machinery is a complicated problem. In this paper, we present an approach for computing all non-homotopic paths characteristic of all the homotopy classes associated with a pair of source–destination points embedded on the surface of the product. We introduce the notion of routing graphs, which are generated from specific non-trivial loops in the first homology group associated with the surface of the product. The routing graph, which is constructed only once, encodes the homotopy of the routing environment and is, therefore, used to find all the non-homotopic paths between multiple source–destination pairs. Based on the path embedding requirements, we classify the routing process into (1) On-Surface Routing, when the paths are constrained to the surface of the product, and (2) In-Air Routing, when the paths are allowed to venture into the product ambiance space. In the former case, the routing graph is defined by both the handle and tunnel loops in the first homology group, whereas in the latter case, it is defined by only the handle loops. We propose a linking number-based strategy for computing the handle and tunnel loops for a given closed surface. The methods are extended for routing in assemblies using the so-called shell structures associated with them. The results obtained demonstrate the efficacy of the proposed approach. The results show that the computation of the routing graph is very efficient, and the homotopy classes can be generated in very little time using the proposed approach.

Proper Equivariant Stable Homotopy Theory

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective ‘proper’ alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective ‘genuine’ indicates that the theory comes with appropriate transfers and Wirthmüller isomorphisms, and the resulting equivariant cohomology theories support the analog of an R O ( G ) R O(G) -grading. Our model for genuine proper G G -equivariant stable homotopy theory is the category of orthogonal G G -spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of G G . This class of π ∗ \pi _* -isomorphisms is part of a symmetric monoidal stable model structure, and the associated tensor triangulated homotopy category is compactly generated. Consequently, every orthogonal G G -spectrum represents an equivariant cohomology theory on the category of G G -spaces. These represented cohomology theories are designed to only depend on the ‘proper G G -homotopy type’, tested by fixed points under all compact subgroups. An important special case of our theory are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties in the sense of geometric group theory; for example, the G G -sphere spectrum is a compact object in our triangulated equivariant homotopy category if the universal space for proper G G -actions has a finite G G -CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper G G -CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by G G -vector bundles. Via this description, we can identify the previously defined G G -cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal G G -spectra.

Minimizing immersions of a hyperbolic surface in a hyperbolic 3-manifold

abstract: Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian $3$-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These ``minimizing'' maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps---but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in $M$. We prove the uniqueness of smooth minimizing maps from $(S,h)$ to $M$ in a given homotopy class. When $(S,h)$ is fixed, smooth minimizing maps from $(S,h)$ are described by a simple holomorphic data on $S$: a complex self-adjoint Codazzi tensor of determinant $1$. The space of admissible data is smooth and naturally equipped with a complex structure, for which the monodromy map taking a data to the monodromy representation of the image is holomorphic. Minimizing maps are in this way reminiscent of shear-bend coordinates, with the complexification of $F$ analoguous to the complex length.

Dynamic recrystallization under hot deformation of additively manufactured 316 L stainless steel

The high-temperature deformation behavior of additively manufactured 316 L stainless steels prepared via the laser powder bed fusion (LPBF) technology was investigated using Gleeble compression testing in a wide temperature range of 473–1273 K under a constant strain rate of 0.001 s−1, perpendicular to the building direction (BD). The softening flow property upon continuous concurrent isothermal heating deformation displayed an excellent thermal stability performance up to the temperature of 873 K, preserving the material strength higher than 600 MPa compared to the room-temperature property of the as-built material. By the gradual increase of testing temperature, the compressive strength of the material was continuously decreased to even <50 MPa at a temperature of 1273 K. After hot compression; all tested specimens were characterized across different regions in terms of deformation substructure evolution and crystallographic texture using electron backscattering diffraction (EBSD) and constitutive analyses. Accordingly, the estimated activation energies based on established constitutive modeling proposed the operation of dynamic recovery (DRV) and discontinuous dynamic recrystallization (DDRX) mechanisms for the hot deformation of additively manufactured 316 L stainless steel below (~9.77kJmol) and higher (~553.23kJmol) than the critical temperature of 873 K, respectively. The competition between strain accumulation and thermal heating induced two different Brass deformation and Goss recrystallization textures below and above this critical temperature by altering the operative dominant dynamic restoration mechanism, with various texture severity depending on the testing temperature.

On the swap move of letters in words and the forbidden moves in virtual diagrams

In the theory of topology of words, the move to swap two adjacent letters in words is a simple operator that makes an [Formula: see text]-homotopy class of a nanoword into the [Formula: see text]-homotopy class of the empty word. In this paper, we study the relation between the swap move of letters and the forbidden moves of virtual diagrams, that is, local moves of virtual diagrams which unknot any virtual knot.

Monitoring artificial canals with multiple SAR satellites: A case study of the Changge Canal of the South-to-North Water Diversion Project in China

Surface monitoring of artificial canals is essential to ensure safe operation. Traditional geodetic tools are commonly employed to monitor the canal of China’s South-to-North Water Diversion Project (SNWDP), which is time-consuming and high-cost. The advanced satellite Synthetic Aperture Radar Interferometry (InSAR) is an effective tool to detect subtle surface deformation, while its application for monitoring canals is short of fine deformation interpretation on a structure scale. In this study, we applied multiple SAR satellites to monitor a typical canal of the Middle Route of the SNWD Project (MRP), the Changge Canal, by extracting the historical deformation, inverting two-dimensional (2D) deformation, and analyzing distortions. The result suggests that the focused canal section is experiencing continuous deformation caused by near coal mining activity. Its vertical and horizontal distortions both reach ±1.5 mm/100 m. The InSAR applicability, including visibility and sensitivity, is applied to the structure-level interpretation of deformation. The InSAR MPs distribution highly depends on canal structures' aspect and surface coverings. The canal structures with high coherence and lower R-Index (i.e., foreshortening) are favorable for detecting InSAR MPs; meanwhile, a few MPs are detected for the structures with high coherence but high R-index. For the sensitivity of the deformation vector along the downslope, its dependency on the aspect opposes InSAR MPs distribution. The structure-level interpretation considering InSAR applicability helps deploy targeted maintenance and analyze deformation mechanisms. The procedure of fine InSAR monitoring applies to similar water diversion projects.

Gate-Tunable Lifshitz Transition of Fermi Arcs and Its Transport Signatures

One hallmark of Weyl semimetals is the emergence of Fermi arcs (FAs) in surface Brillouin zones, where FAs connect the projected Weyl nodes of opposite chiralities. Unclosed FAs can give rise to various exotic effects that have attracted tremendous research interest. Configurations of FAs are usually thought to be determined fully by the band topology of the bulk states, which seems impossible to manipulate. Here, we show that FAs can be simply modified by a surface gate voltage. Because the penetration length of the surface states depends on the in-plane momentum, a surface gate voltage induces an effective energy dispersion. As a result, a continuous deformation of the surface band can be implemented by tuning the surface gate voltage. In particular, as the saddle point of the surface band meets the Fermi energy, the topological Lifshitz transition takes place for the FAs, during which the Weyl nodes switch their partners connected by the FAs. Accordingly, the magnetic Weyl orbits composed of the FAs on opposite surfaces and chiral Landau bands inside the bulk change their configurations. We show that such an effect can be probed by the transport measurements in a magnetic field, in which the switch-on and switch-off conductances by the surface gate voltage signal the Lifshitz transition. Our work opens a new route for manipulating the FAs by surface gates and exploring novel transport phenomena associated with the topological Lifshitz transition.

Slope Deformation Associated With Recent Tectonism and the Lasting Effect of Local Subsurface Geometry in the Taurus‐Littrow Valley, Apollo 17 Landing Site

AbstractThe South Massif and Taurus‐Littrow valley represent a unique area for understanding recent geological processes on the Moon. The presence of two recent overlapping landslide deposits, and boulder falls, suggests that repetitive instability has affected the north‐east facing slope of the South Massif. The presence of the young Lee‐Lincoln lobate scarp associated with a thrust fault suggests that seismic shaking may have been an important factor in triggering surface changes and mass‐wasting events in the area. In this work, we use the younger landslide deposit as a geomorphological marker. The age of the deposit, 70–110 Ma, is known due to the returned samples of the Apollo 17 mission, therefore allowing to set a time constraint to surface changes that have occurred since its emplacement. Here, we extend the body of evidence of slope deformation of the north‐east slope of the South Massif post‐dating the emplacement of the younger landslide deposit. We map boulder tracks, zones of disturbed regolith, summit and slope structures. We described their mutual relationships and their relationships with the topography and local tectonic structures. We identified features directly related to the local stress field, as well as features derived from gravitational adjustment following basal slope support removal due to reactivation of the ancient valley‐bounding fault in reverse mode associated with the Lee‐Lincoln thrust fault. Our interpretation favors a scenario in which recent tectonism, coupled with long‐lasting influence of the subsurface geometry, has caused continuous slope deformation of the South Massif.

Computing Homotopy Classes for Diagrams

We present an algorithm that, given finite diagrams of simplicial sets X, A, Y, i.e., functors {mathcal {I}}^textrm{op}rightarrow {textsf {s}} {textsf {Set}}, such that (X, A) is a cellular pair, dim Xle 2cdot {text {conn}}Y, {text {conn}}Yge 1, computes the set [X,Y]^A of homotopy classes of maps of diagrams ell :Xrightarrow Y extending a given f:Arightarrow Y. For fixed n=dim X, the running time of the algorithm is polynomial. When the stability condition is dropped, the problem is known to be undecidable. Using Elmendorf’s theorem, we deduce an algorithm that, given finite simplicial sets X, A, Y with an action of a finite group G, computes the set [X,Y]^A_G of homotopy classes of equivariant maps ell :Xrightarrow Y extending a given equivariant map f:Arightarrow Y under the stability assumption dim X^Hle 2cdot {text {conn}}Y^H and {text {conn}}Y^Hge 1, for all subgroups Hle G. Again, for fixed n=dim X, the algorithm runs in polynomial time. We further apply our results to Tverberg-type problem in computational topology: Given a k-dimensional simplicial complex K, is there a map Krightarrow {mathbb {R}}^d without r-tuple intersection points? In the metastable range of dimensions, rdge (r+1) k+3, the problem is shown algorithmically decidable in polynomial time when k, d, and r are fixed.

The Third Cohomology 2-Group

In this paper we show that a finite product preserving opfibration can be factorized through an opfibration with the same property, but with groupoidal fibres. If moreover the codomain is additive, one can endow each fibre of the new opfibration with a canonical symmetric 2-group structure. We then apply such factorization to the opfibration that sends a crossed extension of a group C to its corresponding C-module. The symmetric 2-group structure so obtained on the fibres, defines the third cohomology 2-group of C, with coefficients in a C-module. We show that the usual third and second cohomology groups are recovered as its homotopy invariants. Furthermore, even if all results are presented in the category of groups, their proofs are valid in any strongly protomodular semi-abelian category, once one adopts the corresponding internal notions.

Fabrication of Insoluble Elastin by Enzyme-Free Cross-Linking.

Elastin is an essential extracellular matrix protein that enables tissues and organs such as arteries, lungs, and skin, which undergo continuous deformation, to stretch and recoil. Here, we describe an approach to fabricate artificial elastin with close-to-native molecular and mechanical characteristics. We polymerized recombinantly produced tropoelastin through coacervation and allysine-mediated cross-linking induced by pyrroloquinoline quinone (PQQ). We developed a technique that allows the recovery and repeated use of PQQ for protein cross-linking by covalent attachment to magnetic Sepharose beads. The produced material closely resembles natural elastin in its molecular, biochemical and mechanical properties, enabled by the occurrence of the cross-linking amino acids desmosine, isodesmosine and merodesmosine. It possesses elevated resistance against tryptic proteolysis and its Young's modulus ranging between 1-2MPa is similar to that of natural elastin. The approach described herein enables the engineering of mechanically resilient, elastin-like materials for biomedical applications. This article is protected by copyright. All rights reserved.

Nonlocal geometric fracture theory: A nonlocal‐local asymptotically compatible framework for continuous–discontinuous deformation of solid materials

SummaryOne of the fundamental but unsolved problems in fracture mechanics of solid materials is how to describe the continuous and discontinuous deformation of continuum in a unified manner. This paper aims to develop a novel nonlocal geometric fracture theory attempting to address this essential issue. Using the concepts of nonlocal calculus, a new nonlocal equation for the deformation of continuum is obtained by minimizing a nonlocal Lagrangian. This is done to ensure that the proposed nonlocal model can be used to describe the continuous and discontinuous deformation of a continuum in a unified geometric manner, as well as to be asymptotically compatible with the classical continuum theory. Inspired by the phase field fracture theory, a nonlocal geometric crack surface functional is proposed to approximate the Griffith fracture energy in continuum. Note, however, that a distinguished feature of the proposed functional is that it enables strong discontinuities appear in the crack phase fields, which is difficult for the phase field fracture theory. After obtaining the nonlocal fracture energy, a nonlocal elastic energy is phenomenologically decomposed into volumetric and deviatoric parts, and then a system of nonlocal geometric fracture equations is derived to describe the damage nucleation, cracks initiation and propagation, which covers the whole physical process of fracture in solid materials. A staggered discontinuous Galerkin method is developed to reach an accurate and stable numerical solution. Numerical examples demonstrate the proposed theory can well capture the transition from the continuous to discontinuous deformation of the continuum and accurately describe the topological evolution of cracks in a unified framework.

Coarse Sheaf Cohomology

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0, they see the number of ends of the space. In this paper, a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition, coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus, we can use topological tools on compact Hausdorff spaces in our computations. In particular, if the asymptotic dimension of a proper metric space is finite, then higher cohomology groups vanish. We compute a few examples. As it turns out, finite abelian groups are best suited as coefficients on finitely generated groups.

Mechanical responses of anchoring structure under triaxial cyclic loading

Dynamic load on anchoring structures (AS) within deep roadways can result in cumulative damage and failure. This study develops an experimental device designed to test AS under triaxial loads. The device enables the investigation of the mechanical response, failure mode, instability assessment criteria, and anchorage effect of AS subjected to combined cyclic dynamic-static triaxial stress paths. The results show that the peak bearing strength is positively correlated with the anchoring matrix strength, anchorage length, and edgewise compressive strength. The bearing capacity decreases significantly when the anchorage direction is severely inclined. The free face failure modes are typically transverse cracking, concave fracturing, V-shaped slipping and detachment, and spallation detachment. Besides, when the anchoring matrix strength and the anchorage length decrease while the edgewise compressive strength, loading rate, and anchorage inclination angle increase, the failure intensity rises. Instability is determined by a negative tangent modulus of the displacement-strength curve or the continued deformation increase against the general downward trend. Under cyclic loads, the driving force that breaks the rock mass along the normal vector and the rigidity of the AS are the two factors that determine roadway stability. Finally, a control measure for surrounding rock stability is proposed to reduce the internal driving force via a pressure relief method and improve the rigidity of the AS by full-length anchorage and grouting modification.

Reconstruction of spatially continuous time-series land subsidence based on PS-InSAR and improved MLS-SVR in Beijing Plain area

ABSTRACT Beijing has undergone severe settlement in recent years. Persistent Scatterers Interferometric Synthetic Aperture Radar (PS-InSAR) technique has been widely used to derive time-series land deformation. However, existing studies have faced two challenges: (1) the nonlinear characteristics of time-series subsidence has not been fully investigated; (2) since PS points are normally distributed in urban areas with high building density, measurement gaps usually exist in nonurban areas. To address the challenges, we presented a new method to reconstruct spatially continuous time-series deformation. First, PS-InSAR was used to retrieve the deformation based on 135 scenes of Envisat ASAR and Radarsat-2 images from 2003 to 2020. Polynomial Curve Fitting (PCF) was then used to model nonlinear time-series deformation for the PS points. In the PS measurement gaps, Iterative Self-Organizing Data Analysis Technique (ISODATA) and Multi-output Least Squares Support Vector Regression (MLS-SVR) were used to estimate the PCF coefficients and then time-series deformation considering 40 features including thickness of the compressible layers, annual groundwater level, etc. The major results showed that (1) compared to linear, quadratic, and quartic models, cubic polynomial model generated better fit for the time-series deformation (R2 ≈0.99), suggesting obvious nonlinear temporal pattern of deformation; (2) the time-series deformation over measurement gaps reconstructed by ISODATA and MLS-SVR had satisfactory accuracy (R2 = 0.92, MAPE < 15%) and yielded higher accuracy (R2 = 0.947) than IDW (R2 = 0.687) and Ordinary Kriging (R2 = 0.688) interpolation methods. The reconstructed results maintain the nonlinear characteristics and ensure the high spatial resolution (120 m) of time-series deformation. Among the 40 predictor variables, ground water level datasets are the most influential predictors of time-series deformation.

Equivariant algebraic K-theory, G-theory and derived completions

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him, and also appearing in the original Atiyah-Segal theorem. The main goal of the present paper is to provide a proof of this conjecture in as broad a context as possible, making use of the technique of derived completion, and to consider several of the applications.Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the equivariant algebraic G-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). Restricting to actions by split tori, we are also able to consider actions on algebraic spaces. Moreover, the restriction that the base scheme be a field is also not required often, but is put in mainly to simplify some of our exposition. These enable us to obtain a wide range of applications, some of which are briefly sketched and which we plan to explore in detail in the future. In fact, we discuss an extension of our results to equivariant homotopy K-theory along with various Riemann-Roch theorems in a sequel. A comparison of our results with previously known results, none of which made use of derived completions, shows that without the use of derived completions one can only obtain results which are indeed very restrictive.