Deformation Space Research Articles

The combinatorics of Farey words and their traces

We introduce a family of 3 -variable ‘Farey polynomials’ that are closely connected with the geometry and topology of 3 -manifolds and orbifolds as they can be used to produce concrete realisations of the boundaries and local coordinates for one-dimensional (over \mathbb{C} ) deformation spaces of Kleinian groups. As such, this family of polynomials has a number of quite remarkable properties. We study these polynomials from an abstract combinatorial viewpoint, including a recursive definition extending that which is known in the literature for the special case of manifolds, even beyond what the geometry predicts. We also present some intriguing examples and conjectures which we would like to bring to the attention of researchers interested in algebraic combinatorics and hypergeometric functions. The results in this paper additionally provide a practical approach to various classification problems for rank 2 subgroups of \operatorname{PSL}(2,\mathbb{C}) since they, together with other recent work of the authors, make it possible to provide certificates that certain groups are discrete and free, and effective ways to identify relators.

Entropies, level-density parameters and fission probabilities along the triaxially- and axially-symmetric fission paths in 296Lv

We investigate the variation of statistical properties of the fissile nucleus, especially entropy, and nuclear level density, along different fission paths. The calculations were focused on comparing axial and triaxial trajectories leading to fission of 296Lv. We observe that change of shell effects and their suppression rates with deformation can substantially influence fission dynamics. Furthermore, the fission process exhibits iso-entropic behavior at high excitation energies, while pronounced entropy variations are observed at lower energies. We derive a deformation-dependent level density parameter that plays a critical role in estimating the survival probability of a superheavy nucleus. The competition between different fission paths was further studied by employing a master equation approach, thereby demonstrating the critical role of entropy and thermodynamic properties in shaping fission dynamics within multidimensional deformation spaces.

Ethics and Personality Deformation in Time and Space: A Semiotic Analysis of Edgar Allan Poe’s The Black Cat

Romantic literature rose in the 19th century, and the Gothic novel evolved into one of the special genres. Allan Poe carries out the idea of unity of effect. He uses internal and external perspective conversion in The Black Cat, reflecting the continuity of self-consciousness in different times and spaces. It realizes the interpretation of macroscopic ethics and morality connotation. The study focuses on the linguistic symbolic context. It discusses the changing process of the deformed personality of the black cat from the space-time construction implied in the narrative layer. This discussion has research significance in analyzing the signifier meaning of each language symbol and the relationship network of the ideographic system. The construction of time and space, which is full of philosophical thinking, is introduced into the interpretation of the text. At the same time, it further deepens the understanding of Ellen Poe’s moral aesthetic angle. The literature review method and text analysis method are used in this study. From the perspective of the new combination of philosophy and philology theory, it goes further into the inner world of the characters. With the outburst of the negative suggestion power of the protagonist, the deformity of the character also reaches the climax of the intensity. From an interdisciplinary perspective, literary criticism offers a new way to the Alienation of capitalist society through the involuntary flow of consciousness and the conversion of narrative layers.

Precessing non-axisymmetric ellipsoids: bi-stability and fluid instabilities

This study explores precession-driven flows in a non-axisymmetric ellipsoid spinning around its medium axis. Using an experimental approach, we focus on two aspects of the flow: the base flow of uniform vorticity and the development of fluid instabilities. In contrast to a preceding paper (J. Fluid. Mech., vol. 932, 2022, A24), where the ellipsoid rotated around its shortest axis, we do not observe bi-stability or hysteresis of the base flow, but a continuous transition from small to large differential rotation and tilt of the fluid rotation axis. We then use the model developed by Noir & Cébron (J. Fluid. Mech., vol. 737, 2013, pp. 412–439) to numerically determine regions in the parameter space of axial and equatorial deformations for which bi-stability may exist. Concerning fluid instabilities, we use three independent observations to track the onset of both boundary layer and parametric instabilities. Our results clearly show the presence of a parametric instability, yet the exact nature of the underlying mechanism (conical shear layer instability, shear instability and elliptical instability) is not unambiguously identified. A coexisting boundary layer instability, although unlikely, cannot be ruled out based on our experimental data. To make further progress on this topic, a new generation of experiments at significantly lower Ekman numbers (ratio of rotation and viscous time scales) is clearly needed.

A new model describing improved bound-state and statistical properties in the framework of deformed Schrödinger equation for the improved Coshine Yukawa potential in 3D-NR(NCPS) symmetries

In this paper, within the three-dimensional non-relativistic non-commutative phase–space (3D-NR(NCPS)) symmetries, we examined the 3D deformed Schrödinger equation (3D-DSE) using the improved Coshine Yukawa potential (ICYP) model composed from Coshine Yukawa potential (CYP) ([Formula: see text]) and other terms produced from the effect of phase–space deformation ([Formula: see text] and [Formula: see text]). For this study, the 3D-DSE in the 3D-NR(NCPS) symmetry is solved and discussed with standard independent time perturbation theory and the generalized Bopp’s shifts method. For the homogeneous (H2 and N2) and heterogeneous (LiH, ScH, and HCL) diatomic molecules, it is obtained that the improved non-relativistic energy equation and eigenfunction for the ICYP in the presence of deformation phase–space are dependent on the discrete atomic quantum numbers ([Formula: see text],m), the dissociation energy [Formula: see text], the equilibrium bond length the screening parameter [Formula: see text], the deformation phase parameters ([Formula: see text]) and the deformation space parameters ([Formula: see text]). Additionally, the thermal properties of the CYP and ICYP with 3D Schrödinger equation (3D-SE) and 3D-DSE are thoroughly examined in the three-dimensional non-relativistic quantum mechanics (3D-NR(QM)) known in the literature symmetry and 3D-NR(NCPS) symmetries, including the partition function, mean energy, free energy, specific heat, and entropy. Furthermore, we discussed particular cases of thermodynamic characteristics for the ICYP model. In our study, we have shown that all the physical values related to energy and thermodynamic properties within the framework of the 3D(NR)NCPS symmetry are equal to the corresponding values within the framework of 3D-NR(QM) symmetry known in the literature, in addition to minor effects resulting from their interaction with the topological properties of the deformed phase–space. This study has multiple applications in various domains, including atomic and molecular physics.

Automorphism Groups of Deformations and Quantizations of Kleinian Singularities

AbstractIt is known that, for the algebra of functions on a Kleinian singularity, the parameter space of deformations and the parameter space of quantizations coincide. We prove that, for a Kleinian singularity of type $$\textbf{A}$$ A or $$\textbf{D}$$ D , isomorphisms between the quantizations are essentially the same as Poisson isomorphisms between deformations. In particular, the group of automorphisms of the deformation and the quantization corresponding to the same deformation parameter are isomorphic. We additionally describe the groups of automorphisms as abstract groups: for type $$\textbf{A}$$ A they have an amalgamated free product structure, for type $$\textbf{D}$$ D they are subgroups of the groups of Dynkin diagram automorphisms. For type $$\textbf{D}$$ D we also compute all the possible affine isomorphisms between deformations; this was not known before.

Technology and application of through layer borehole ultra-high pressure hydraulic slotting in soft coal seam to pressure relief and permeability enhancement

ABSTRACT It is a vital method that controls outburst disasters through gas drainage to reduce gas pressure. But the permeability of coal seam in China is low. Therefore, technologies that can decrease pressure and increase permeability such as ultra-high pressure hydraulic slotting (UPHS) are widely applied. Based on the technology of UPHS, the principle of coal breaking and mechanism of pressure relief and permeability enhancement (PRPE) were first analyzed. The effects of PRPE between contrast borehole and UPHS were compared through numerical simulation. Finally, combined with the field test, the parameters of UPHS were determined, and difference in gas drainage effect between contrast borehole and UPHS was investigated. It is found that the exposed area of coal increased due to macroscopic slotting formed by hydraulic slotting, which provide space for coal deformation. The stress of coal relieved the permeability enhanced, resulting in the gas drainage efficiency increasing. The simulated results showed the pressure relief area and plastic deformation caused by UPHS are significantly increased compared with contrast borehole. The average concentration and total of gas drainage in UPHS are 1.49 and 3.13 times than that of contrast borehole. And radius of gas drainage of UPHS 1.77 times than that of contrast borehole under reduce the 46% engineering quantities.

Towards a modular construction of OG10

We construct the first example of a stable hyperholomorphic vector bundle of rank five on every hyper-Kähler manifold of $\mathrm {K3}^{[2]}$ -type whose deformation space is smooth of dimension 10. Its moduli space is birational to a hyper-Kähler manifold of type OG10. This provides evidence for the expectation that moduli spaces of sheaves on a hyper-Kähler could lead to new examples of hyper-Kähler manifolds.

Development of a Novel Beam-Based Finite-Element Approach for the Computationally Efficient Prediction of Residual Stresses and Displacements in Large 3D-Printed Polymer Parts

Recently, 3D printing of large, structural polymer parts has received increasing interest, especially for the creation of recyclable structural parts and tooling. However, the complexity of large-scale 3D polymeric printing often dictates resource-intensive trial and error processes to achieve acceptable parts. Existing computational models used to assess the impact of fabrication conditions typically treat the 3D-printed part as a continuum, incorporate oversimplified boundary conditions and take hours to days to run, making design space exploration infeasible. The purpose of this study is to create a structural model that is computationally efficient compared with traditional continuum models yet retains sufficient accuracy to enable exploration of the design space and prediction of part residual stresses and deformations. To this end, a beam-based finite element methodology was created where beads are represented as beams, vertical springs represent inter-bead transverse force transfer and multi-point, linear constraints enforce strain compatibility between adjacent beads. To test this framework, the fabrication of a large Polyethylene terephthalate glycol (PETG) wall was simulated. The PETG was modeled as linearly elastic with an experimentally derived temperature-dependent coefficient of thermal expansion and elastic modulus using temperature history imported from an ABAQUS thermal model. The results of the simulation were compared to those from a continuum model with an identical material definition, showing reasonable agreement of stresses and displacements. Further, the beam-based model required an order of magnitude less run time. Subsequently, the beam-based model was extended to allow separation of the part from the printing bed and the inclusion of part self-weight during fabrication to assess the significance of these effects that pose challenges for existing continuum models.

Divisible convex sets with properly embedded cones

AbstractIn this article we construct many examples of properly convex irreducible domains divided by Zariski dense relatively hyperbolic groups in every dimension at least 3. This answers a question of Benoist. Relative hyperbolicity and non-strict convexity are captured by a family of properly embedded cones (convex hulls of points and ellipsoids) in the domain. Our construction is most flexible in dimension 3 where we give a purely topological criterion for the existence of a large deformation space of geometrically controlled convex projective structures with totally geodesic boundary on a compact 3-manifold.

Failure mechanism of severe deformation in gob-side entry rib and unloading-supporting coupling control technology

Given the problem of the solid coal side produces strong mine pressure in the stage of strong mining and lagging dynamic pressure in the gob-side entry retaining (GER). Using the W3233 ventilation roadway in the West-Third mining area as a case study, the research employs numerical simulation, field observation, and engineering analogy. Key findings include: (1) The optimal plan reduces maximum stress on the coal side and shifts it inward, primarily by severing the main roof and alleviating the gangue wall load, offering limited pressure relief on the solid coal rib. (2) When hole-creating depth is too close to the solid coal rib (b = 6 m), it compromises integrity. If the depth exceeds the stress maximum region (b = 10 m), it fails to effectively sever the peak stress area. However, a depth within the stress maximum region (b = 8 m) effectively absorbs and transfers stress, with a buffer area of approximately 3.7 m from the external support area. (3) The length of the hole (l = 2, 4, 6 m) directly affects deformation absorption and pressure relief, while longitudinal row spacing (s = 2.7, 3.6, 4.5 m) has an inverse effect. For optimal results, a hole length of 4 m and a row spacing of 3.6 m are recommended. (4) After 150 days, the new bag pressure regulating system maintains about 40 % of the absorbable deformation space in the LPRH, confirming simulation results. Observations show that external support forces remain stable at around 250 kN, and coal rib displacement is controlled within 350 mm, over 65 % less than similar GER sections with comparable geological conditions. This research introduces a novel technology and methodology for managing large deformations of the solid coal rib in gob-side dynamic pressure roadways, providing significant reference value.

The category $\Theta_{2}$, derived modifications, and deformation theory of monoidal categories

A complex C^{\bullet}(C,D)(F,G)(\eta,\theta) , generalising the Davydov–Yetter complex of a monoidal category (Davydov (1998) and Yetter (1998)), is constructed. Here, C,D are \Bbbk -linear (corresp., dg) bicategories, F,G\colon C\to D are \Bbbk -linear (corresp., dg) strong functors, and \eta,\theta\colon F\Rightarrow G are strong natural transformations. Morally, it is a complex of “derived modifications” \eta\Rrightarrow\theta ; likewise for the case of dg categories, one has the complex of “derived natural transformations” F\Rightarrow G , given by the Hochschild cochain complex of C with coefficients in C -bimodule D(F-,G=) .The complex C^{\bullet}(C,D)(F,G)(\eta,\theta) naturally arises from a 2-cocellular dg vector space A(C,D)(F,G)(\eta,\theta)\colon\Theta_{2}\to C^{\bullet}(\Bbbk) , as its \Theta_{2} -totalisation (here, \Theta_{2} is the category dual to the category of Joyal 2-disks (Joyal (1997))).It is shown that for a \Bbbk -linear monoidal category C , the third cohomology vector space H^{3}(C^{\bullet}(C,C)(\mathrm{Id},\mathrm{Id})(\mathrm{id},\mathrm{id})) is isomorphic to the vector space of the outer (modulo twists) infinitesimal deformations of the \Bbbk -linear monoidal category which we call the full deformations. It means that the following data is to be deformed: (a) the underlying dg category structure, (b) the monoidal product on morphisms (the monoidal product on objects is a set-theoretical datum and is maintained under the deformation), and (c) the associator. The data (a), (b), (c) is subject to the (infinitesimal versions of) numerous monoidal compatibilities, which we interpret as the closeness of the corresponding degree 3 element. Similarly, H^{2}(C^{\bullet}(C,D)(F,F)(\mathrm{id},\mathrm{id})) is isomorphic to the vector space of the outer infinitesimal deformations of the strong monoidal functor F .A relative totalisation Rp_{*}A(C,D)(F,F)(\mathrm{id},\mathrm{id}) along the projection p\colon\Theta_{2}\to\Delta is defined, and it is shown to be a cosimplicial monoid, which fulfils the Batanin–Davydov 1-commutativity condition (Batanin and Davydov (2023)). Then it follows from loc. cit. that C^{\bullet}(C,D)(F,F)(\mathrm{id},\mathrm{id}) is a C_{\bullet}(E_{2};\Bbbk) -algebra. Conjecturally, C^{\bullet}(C,C)(\mathrm{Id},\mathrm{Id})(\mathrm{id},\mathrm{id}) is a C_{\bullet}(E_{3};\Bbbk) -algebra; however, the proof requires more sophisticated methods.

Extremal weights and a tameness criterion for mod $p$ Galois representations

We study the weight part of Serre’s conjecture for generic n -dimensional mod p Galois representations. We first generalize Herzig’s conjecture to the case where the field is ramified at p and prove the weight elimination direction of the conjecture. We then introduce a new class of weights associated to n -dimensional local mod p representations which we call extremal weights . Using a “Levi reduction” property of certain potentially crystalline Galois deformation spaces, we prove the modularity of these weights. As a consequence, we deduce the weight part of Serre’s conjecture for unit groups of some division algebras in generic situations.

Classical Liouville action and uniformization of orbifold Riemann surfaces

We study the classical Liouville field theory on Riemann surfaces of genus g>1 in the presence of vertex operators associated with branch points of orders mi>1. In order to do so, we will consider the generalized Schottky space Sg,n(m) obtained as a holomorphic fibration over the Schottky space Sg of the (compactified) underlying Riemann surface. The fibers of Sg,n(m)→Sg correspond to configuration spaces of n orbifold points of orders m=(m1,…,mn). Drawing on the previous work of Park [] as well as Takhtajan and Zograf [; L. A. Takhtajan and P. Zograf], we define Hermitian metrics hi for tautological line bundles ℒi over Sg,n(m). These metrics are expressed in terms of the first coefficient of the expansion of covering map J near each singular point on the Schottky domain. Additionally, we define the regularized classical Liouville action Sm using Schottky global coordinates on Riemann orbisurfaces with genus g>1. We demonstrate that exp[Sm/π] serves as a Hermitian metric in the holomorphic Q-line bundle ℒ=⊗i=1nℒi⊗(1−1/mi2) over Sg,n(m). Furthermore, we explicitly compute the first and second variations of the smooth real-valued function 𝒮m=Sm−π∑i=1n(mi−1mi)loghi on the Schottky deformation space Sg,n(m). We establish two key results: (i) 𝒮m generates a combination of accessory and auxiliary parameters, and (ii) −𝒮m acts as a Kähler potential for a specific combination of Weil-Petersson and Takhtajan-Zograf metrics that appear in the local index theorem for orbifold Riemann surfaces [Takhtajan and Zograf, ]. The obtained results can then be interpreted in terms of the complex geometry of the Hodge line bundle equipped with Quillen’s metric over the moduli space Mg,n(m) of Riemann orbisurfaces and the tree-level approximation of conformal Ward identities associated with quantum Liouville theory. Published by the American Physical Society 2024

New Solutions of Lorentz Transformation IV

This paper is the fourth part of a hypothesis originally based on the basic assumptions of Lorentz transformation and has various implications. In the first part of the hypothesis [1], I calculated the wave function from the general assumptions of the Lorentz transformation. This wave function describes spacetime deformations and entirely replaces the original Lorentz solution used in special relativity. Importantly, each new solution, for both time and space deformation, has two possible solutions that are equally probable. Therefore, I have used these equations for further calculations, which already have a quantum nature. In the second part of my hypothesis [1], I converted this equation into an electromagnetic one and used it to calculate interference and diffraction. Thus, the resulting equation is not based on complex functions, as in standard calculations. We can further investigate this equation, for example, in the context of electron levels in an atom, as interference and diffraction are phenomena related to Young's experiment, and the wave properties of electrons have been demonstrated. In the third part of my hypothesis [1], I applied the calculations to atomic relations and outlined possible solutions for atomic orbitals. This outline of the potential arrangement of energies in the atomic model arose from the fact that some molecules, such as CH4, have the shape of a Platonic solid tetrahedron, which I consider pivotal within the framework of the VSEPR theory.

Inscribable fans I: Inscribed cones and virtual polytopes

AbstractWe investigate polytopes inscribed into a sphere that are normally equivalent (or strongly isomorphic) to a given polytope . We show that the associated space of polytopes, called the inscribed cone of , is closed under Minkowski addition. Inscribed cones are interpreted as type cones of ideal hyperbolic polytopes and as deformation spaces of Delaunay subdivisions. In particular, testing if there is an inscribed polytope normally equivalent to is polynomial time solvable. Normal equivalence is decided on the level of normal fans and we study the structure of inscribed cones for various classes of polytopes and fans, including simple, simplicial, and even. We classify (virtually) inscribable fans in dimension 2 as well as inscribable permutahedra and nestohedra. A second goal of the paper is to introduce inscribed virtual polytopes. Polytopes with a fixed normal fan form a monoid with respect to Minkowski addition and the associated Grothendieck group is called the type space of . Elements of the type space correspond to formal Minkowski differences and are naturally equipped with vertices and hence with a notion of inscribability. We show that inscribed virtual polytopes form a subgroup, which can be nontrivial even if does not have actual inscribed polytopes. We relate inscribed virtual polytopes to routed particle trajectories, that is, piecewise‐linear trajectories of particles in a ball with restricted directions. The state spaces gives rise to connected groupoids generated by reflections, called reflection groupoids. The endomorphism groups of reflection groupoids can be thought of as discrete holonomy groups of the trajectories and we determine when they are reflection groups.

Noncommutative of Klein–Gordon and Schrödinger equations in the background of the improved Hua plus modified Eckart potential model in 3D-(R/NR)NCQS symmetries: Spectrum and thermodynamic properties

The impact of deformation space on the physical characteristics of diverse physics systems has been thoroughly investigated in research papers. In this work, we study the deformed Klein–Gordon equation (DKGE) in the three-dimensional relativistic non-commutative quantum space (3D-RNCQS) regime by using the improved Hua plus modified Eckart potential (IHPMEP) model. For this consideration, the DKGE in the 3D-RNCQS regime is solved using the standard perturbation theory and the well-known Bopp’s shifts method with the Greene–Aldrich approximation to the centrifugal barrier. The new relativistic energy equation and eigenfunction for the IHPMEP in the presence of deformation space-space for the heterogeneous (CO, HF, and NO) and homogeneous (N2, H2, and Li2) diatomic molecules are obtained to be sensitive to the atomic quantum numbers ([Formula: see text] and [Formula: see text]), the mixed potential depths ([Formula: see text], and [Formula: see text]), the inverse of the screening parameter [Formula: see text], and non-commutativity parameters ([Formula: see text], [Formula: see text], and [Formula: see text]). Analysis is performed on the non-relativistic limit of new energy spectra. By appropriately adjusting the combined potential parameters, we analyze the obtained new bound state eigenvalues of the DKGE and deformed Schrödinger equation with the IHPMEP in 3D-NCQS symmetries and obtain the new modified Eckart potential, the modified Hua potential, the modified Morse potential, and the modified Pöschl–Teller potential. Within the framework of the 3D-NRNCQS regime, the homogeneous and heterogeneous composite systems under IHPMEP models are examined. A thorough investigation is carried out into the impact of space-space deformation on the thermal parameters of the IHPMEP, including the partition function, mean energy, free energy, specific heat, and entropy. This work is of a fundamental absorbability nature and pedagogical interest in atomic and molecular physics.

Biharmonic Coordinates and their Derivatives for Triangular 3D Cages

As a natural extension to the harmonic coordinates, the biharmonic coordinates have been found superior for planar shape and image manipulation with an enriched deformation space. However, the 3D biharmonic coordinates and their derivatives have remained unexplored. In this work, we derive closed-form expressions for biharmonic coordinates and their derivatives for 3D triangular cages. The core of our derivation lies in computing the closed-form expressions for the integral of the Euclidean distance over a triangle and its derivatives. The derived 3D biharmonic coordinates not only fill a missing component in methods of generalized barycentric coordinates but also pave the way for various interesting applications in practice, including producing a family of biharmonic deformations, solving variational shape deformations, and even unlocking the closed-form expressions for recently-introduced Somigliana coordinates for both fast and accurate evaluations.

Computational Homogenization for Inverse Design of Surface-based Inflatables

Surface-based inflatables are composed of two thin layers of nearly inextensible sheet material joined together along carefully selected fusing curves. During inflation, pressure forces separate the two sheets to maximize the enclosed volume. The fusing curves restrict this expansion, leading to a spatially varying in-plane contraction and hence metric frustration. The inflated structure settles into a 3D equilibrium that balances pressure forces with the internal elastic forces of the sheets. We present a computational framework for analyzing and designing surface-based inflatable structures with arbitrary fusing patterns. Our approach employs numerical homogenization to characterize the behavior of parametric families of periodic inflatable patch geometries, which can then be combined to tessellate the sheet with smoothly varying patterns. We propose a novel parametrization of the underlying deformation space that allows accurate, efficient, and systematical analysis of the stretching and bending behavior of inflated patches with potentially open boundaries. We apply our homogenization algorithm to create a database of geometrically diverse fusing patterns spanning a wide range of material properties and deformation characteristics. This database is employed in an inverse design algorithm that solves for fusing curves to best approximate a given input target surface. Local patches are selected and blended to form a global network of curves based on a geometric flattening algorithm. These fusing curves are then further optimized to minimize the distance of the deployed structure to target surface. We show that this approach offers greater flexibility to approximate given target geometries compared to previous work while significantly improving structural performance.

Analysis of an initially stressed functionally graded thermoelastic medium (type III) without energy dissipation

The current investigation deals with the deformation of a non-homogeneous thermoelastic half space under hydrostatic initial stress for the Green–Naghdi model III. The medium is supposed to be rotating with a constant angular velocity. The non-homogeneous properties of the material are along the x-direction. At the first instance, the problem has been solved analytically to obtain stress and displacement components. Further, the numerical values of these expressions are evaluated using a computer program for a particular medium. The numerical values obtained are then presented graphically to show the effect of initial stress parameter and non-homogeneity parameter on the quantities.