topology-optimization-in-engineering

My question is really just about one-dimensional spaces. I am primarily using covering dimension but I'd be sufficiently happy with an answer for small or large inductive dimension. Question: Suppose ...

This paper introduces the Signature Dynamics framework, a discrete dynamical system that algorithmically generates metallic signatures for Diophantine approximation by the Universal Metallic Family α(n, N) = n + √(n² + N). For a target constant T > 0, the map Φ_T(n) = round(T² · n) generates an orbit L(T) whose consecutive terms (n_{m+1}, n_m) are, by construction, valid metallic signatures of T.…

This is a question from Page 133 of George F. Simmons’ ‘Introduction to Topology and Modern Analysis’. Let $X$ be a topological space and $Y$ a Hausdorff space and $A$ a subspace of $X$. Show that a ...

topology (point-set topology, point-free topology) see also differential topology, algebraic topology, functional analysis and topological homotopy theory Basic concepts fiber space, space attachment Kolmogorov space, Hausdorff space, regular space, normal space sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact Examples B…

Andrée Charles Ehresmann (née Bastiani) is a category theorist. She is editor in chief of the journal Cahiers de Topologie et Géométrie Différentielle Catégoriques, founded by her late husband Charles Ehresmann. Andrée Ehresmann’s early papers were published under her maiden name Andrée Bastiani, and during a transition period under the hyphenated surname Bastiani-Ehresmann. On internalization of…

∞-Lie theory (higher geometry) Background Smooth structure Higher groupoids Lie theory ∞-Lie groupoids ∞-Lie algebroids Formal Lie groupoids Cohomology Homotopy Related topics Examples -Lie groupoids -Lie groups -Lie algebroids -Lie algebras A Lie 2-algebra is to a Lie 2-group as a Lie algebra is to a Lie group. Thus, it is a vertical categorification of a Lie algebra. A (“semistrict”) Lie 2-alge…

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … topology (point-set topology, point-free topology) see also differential topology, algebraic topology, functional analysis and topological homotopy theory Basic concepts fiber space…

analysis (differential/integral calculus, functional analysis, topology) metric space, normed vector space open ball, open subset, neighbourhood convergence, limit of a sequence compactness, sequential compactness … … Epsilontic analysis is the now standard rigorous approach to analysis, developed by Bolzano 1817, Augustin Cauchy, Karl Weierstraß, Richard Dedekind, and others in the mid-to-late 1…

Bernard Bolzano (1789-1848) was a Bohemian mathematician, logician, and philosopher. In his lifetime he was renowned perhaps mainly for his Wissenschaftslehre which puts forth an encompassing theory of science. In mathematics, much of his work became well known only after his death, but he can be regarded as an early proponent of epsilontic analysis and the introduction of rigor in analysis, as f…

topology (point-set topology, point-free topology) see also differential topology, algebraic topology, functional analysis and topological homotopy theory Basic concepts fiber space, space attachment Kolmogorov space, Hausdorff space, regular space, normal space sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact Examples B…

topology (point-set topology, point-free topology) see also differential topology, algebraic topology, functional analysis and topological homotopy theory Basic concepts fiber space, space attachment Kolmogorov space, Hausdorff space, regular space, normal space sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact Examples B…

higher geometry / derived geometry Ingredients Concepts geometric little (∞,1)-toposes geometric big (∞,1)-toposes Constructions Examples derived smooth geometry Theorems A geometry is an (∞,1)-category equipped in a compatible way with the structure of an (∞,1)-site; the structure of an essentially algebraic (∞,1)-theory. The objects of are to be thought of as test-spaces with certain higher geo…

On pi-base there was a discussion that Artinian and Noetherian spaces are precisely those spaces that have a finite topology. Say a space is Artinian when its open sets satisfy descending chain ...

topology (point-set topology, point-free topology) see also differential topology, algebraic topology, functional analysis and topological homotopy theory Basic concepts fiber space, space attachment Kolmogorov space, Hausdorff space, regular space, normal space sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact Examples B…

In this little note I offer a topological sketch of one feature (hardly the only one!) of the liar paradox. One may think of the liar as a point lying on the boundary between truth and falsity, allowing for a local entanglement of opposite truth values even while preserving their global distinction. Although the model may generalize to other paradoxes, I make no attempt to do so.

The recent synthesis of triple-pore covalent organic frameworks (TP-COFs), specifically TP-COF-DAB and TP-COF-BZ, represents a significant milestone in framework chemistry, as replicating such multi-pore architectures remains highly challenging. Despite their structural novelty, the mathematical perspective of their topology and associated properties is still limited. In this study, closed-form e…