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"Angles (geometry)"
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Degree Spectra of Relations on a Cone
Let \\mathcal A be a mathematical structure with an additional relation R. The author is interested in the degree spectrum of R, either among computable copies of \\mathcal A when (\\mathcal A,R) is a \"natural\" structure, or (to make this rigorous) among copies of (\\mathcal A,R) computable in a large degree d. He introduces the partial order of degree spectra on a cone and begin the study of these objects. Using a result of Harizanov--that, assuming an effectiveness condition on \\mathcal A and R, if R is not intrinsically computable, then its degree spectrum contains all c.e. degrees--the author shows that there is a minimal non-trivial degree spectrum on a cone, consisting of the c.e. degrees.
eBook
Multi-objective integrated optimization of tool geometry angles and cutting parameters for machining time and energy consumption in NC milling
The manufacturing industry has a large volume and a wide range of energy consumption. It is the main body of energy consumption in the industrial field. In recent years, a great challenge has been posed to the manufacturing industry to improve sustainable development for low energy consumption and high efficiency. The energy efficiency of manufacturing systems has become an important research hotspot in the sustainable development of the world’s manufacturing. Aiming at the problem of low energy efficiency in the current machining process, this paper considers the relationship between tool geometric angles and energy consumption, and on the basis of the current energy consumption model establishes an energy consumption model with independent variables including cutting parameters and tool geometric angles. In order to achieve energy-saving and high efficiency, a multi-objective optimization model was established with cutting parameters and tool geometric angles as optimization variables, combined with actual machine tool and process parameter constraints. The model is solved by the elitist non-dominated sorting genetic algorithm (NSGA-II), and an optimized combination of cutting parameters and tool geometric angle is obtained. Through the comparison experiments conducted with the processing methods that are respectively based on the experience and simply optimize the cutting parameters in the actual milling test, the practivity and validity of the method proposed are verified. The test results show that integrated optimization of cutting parameters and tool geometric angles can reduce energy consumption by 8.77% at least.
Journal Article
Cram Session in Goniometry
When all you need is a basic understanding of goniometry. When concise and illustrative examples of goniometric techniques will provide exactly what the rehabilitation professional needs-Look to
Cram Session in Goniometry: A Handbook for Students and Clinicians
for quick and \"at your fingertips\" facts.
Cram Session in Goniometry
by Lynn Van Ost is a concise and abundantly illustrated quick reference which provides the rehabilitation professional with a very basic approach to various goniometric techniques. Organized in a \"head to toe\" format, Cram Session in Goniometry takes user-friendly and efficient learning to a new level.
What is in your \"Cram Session\":
Photographs depicting each goniometric measurement
Breakdown of each body region covered
Each type of joint, capsular pattern, average range of motion, patient positioning, goniometric alignment, patient substitutions, and alternative methods of measurement
Over 190 photographs
Cram Session in Goniometry: A Handbook for Students and Clinicians
is an informative, well-organized handbook for all students and clinicians in physical therapy, occupational therapy, athletic training and orthopedics.
eBook
Study on high precision calculation method of geometric parameters of blade profile
Two-dimensional blade profile design is the basis of three-dimensional blade modeling. The important parameters reflecting the geometric characteristics of the blade profile are chord length, maximum thickness, inlet and outlet metal angle, installation angle, etc. These parameters are the basis of blade profile design and performance analysis, and it is of great significance to calculate these parameters accurately. Based on the original blade profile data, this paper puts forward a high-precision blade profile parameter extraction method and obtains detailed geometric parameters, including small circle, middle arc and maximum thickness of the blade profile, which lays a foundation for blade profile design and analysis. It is verified that the proposed method has high accuracy and good engineering applicability.
Journal Article
Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization
The 3D Ising transition, the most celebrated and unsolved critical phenomenon in nature, has long been conjectured to have emergent conformal symmetry, similar to the case of the 2D Ising transition. Yet, the emergence of conformal invariance in the 3D Ising transition has rarely been explored directly, mainly due to unavoidable mathematical or conceptual obstructions. Here, we design an innovative way to study the quantum version of the 3D Ising phase transition on spherical geometry, using the “fuzzy (noncommutative) sphere” regularization. We accurately calculate and analyze the energy spectra at the transition, and explicitly demonstrate the state-operator correspondence (i.e., radial quantization), a fingerprint of conformal field theory. In particular, we identify13 parity-even primary operators within a high accuracy and two parity-odd operators that were not known before. Our result directly elucidates the emergent conformal symmetry of the 3D Ising transition, a conjecture made by Polyakov half a century ago. More importantly, our approach opens a new avenue for studying 3D conformal field theories by making use of the state-operator correspondence and spherical geometry.
Journal Article
Hydrogel muscles powering reconfigurable micro-metastructures with wide-spectrum programmability
Stimuli-responsive geometric transformations endow metamaterials with dynamic properties and functionalities. However, using existing transformation mechanisms to program a single geometry to transform into diverse final configurations remains challenging, imposing crucial design restrictions on achieving versatile functionalities. Here, we present a programmable strategy for wide-spectrum reconfigurable micro-metastructures using linearly responsive transparent hydrogels as artificial muscles. Actuated by the hydrogel, the transformation of micro-metastructures arises from the collaborative buckling of their building blocks. Rationally designing the three-dimensional printing parameters and geometry features of the metastructures enables their locally isotropic or anisotropic deformation, allowing controllable wide-spectrum pattern transformation with programmable chirality and optical anisotropy. This reconfiguration mechanism can be applied to various materials with a wide range of mechanical properties. Our strategy enables a thermally reconfigurable printed metalattice with pixel-by-pixel mapping of different printing powers and angles for displaying or hiding complex information, providing opportunities for encryption, miniature robotics, photonics and phononics applications.It is difficult to program a single stimuli-responsive geometry to transform into diverse final configurations in a systematic manner. Here, linearly responsive transparent hydrogels are developed to create micro-metastructures with wide-spectrum thermal reconfigurability.
Journal Article
High-resolution tomographic volumetric additive manufacturing
In tomographic volumetric additive manufacturing, an entire three-dimensional object is simultaneously solidified by irradiating a liquid photopolymer volume from multiple angles with dynamic light patterns. Though tomographic additive manufacturing has the potential to produce complex parts with a higher throughput and a wider range of printable materials than layer-by-layer additive manufacturing, its resolution currently remains limited to 300 µm. Here, we show that a low-étendue illumination system enables the production of high-resolution features. We further demonstrate an integrated feedback system to accurately control the photopolymerization kinetics over the entire build volume and improve the geometric fidelity of the object solidification. Hard and soft centimeter-scale parts are produced in less than 30 seconds with 80 µm positive and 500 µm negative features, thus demonstrating that tomographic additive manufacturing is potentially suitable for the ultrafast fabrication of advanced and functional constructs.
Tomographic additive manufacturing produces complex parts with a wide range of printable materials but remains limited in terms of resolution. Here, the authors tune the étendue of the light source and accurately control the photopolymerization kinetics using an integrated feedback system, leading to the fabrication of high resolution features.
Journal Article
A polyhedron comparison theorem for 3-manifolds with positive scalar curvature
The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is isometric to a flat polyhedron.
Journal Article
Programming shape using kirigami tessellations
Kirigami tessellations, regular planar patterns formed by partially cutting flat, thin sheets, allow compact shapes to morph into open structures with rich geometries and unusual material properties. However, geometric and topological constraints make the design of such structures challenging. Here we pose and solve the inverse problem of determining the number, size and orientation of cuts that enables the deployment of a closed, compact regular kirigami tessellation to conform approximately to any prescribed target shape in two or three dimensions. We first identify the constraints on the lengths and angles of generalized kirigami tessellations that guarantee that their reconfigured face geometries can be contracted from a non-trivial deployed shape to a compact, non-overlapping planar cut pattern. We then encode these conditions into a flexible constrained optimization framework to obtain generalized kirigami patterns derived from various periodic tesselations of the plane that can be deployed into a wide variety of prescribed shapes. A simple mechanical analysis of the resulting structure allows us to determine and control the stability of the deployed state and control the deployment path. Finally, we fabricate physical models that deploy in two and three dimensions to validate this inverse design approach. Altogether, our approach, combining geometry, topology and optimization, highlights the potential for generalized kirigami tessellations as building blocks for shape-morphing mechanical metamaterials.
Journal Article