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In this paper, we construct the super Virasoro algebra with an arbitrary conformal dimension $\mathrm{\Delta }$ from the generalized ${ℛ}(p,q)$-deformed quantum algebra and investigate the ${ℛ}(p,q)$-deformed super Virasoro algebra with the particular conformal dimension $\mathrm{\Delta }=1.$ Furthermore, we perform the ${ℛ}(p,q)$-conformal Virasoro $n$-algebra, the ${ℛ}(p,q)$-conformal super Virasoro $n$-algebra ($n$-even) and discuss a toy model for the ${ℛ}(p,q)$-conformal Virasoro constraints and ${ℛ}(p,q)$-conformal super Virasoro constraints. Besides, we generalized the notion of the ${ℛ}(p,q)$-elliptic Hermitian matrix model with an arbitrary conformal dimension $\mathrm{\Delta }$. Finally, we deduce relevant particular cases generated by quantum algebras known in the literature.

We study knots in 𝕊³ obtained by the intersection of a minimal surface in ℝ⁴ with a small 3-sphere centered at a branch point. We construct new examples of minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or fully amphicheiral; this yields an obstruction for a given knot to be a simple minimal knot. Properties and invariants of these knots such as the algebraic crossing number of a braid representative and the Alexander polynomial are studied.

Carlotto, Chodosh and Rubinstein studied the rate of convergence of the Yamabe flow on a closed (compact without boundary) manifold $M$:

Biconformal spaces contain the essential elements of quantum mechanics, making the independent imposition of quantization unnecessary. Based on three postulates characterizing motion and measurement in biconformal geometry, we derive standard quantum mechanics, and show how the need for probability amplitudes arises from the use of a standard of measurement. Additionally, we show that a postulate for unique, classical motion yields Hamiltonian dynamics with no measurable size changes, while a postulate for probabilistic evolution leads to physical dilatations manifested as measurable phase changes. Our results lead to the Feynman path integral formulation, from which follows the Schrödinger equation. We discuss the Heisenberg uncertainty relation and fundamental canonical commutation relations.

Of those gauge theories of gravity known to be equivalent to general relativity, only the biconformal gauging introduces new structures — the quotient of the conformal group of any pseudo-Euclidean space by its Weyl subgroup always has natural symplectic and metric structures. Using this metric and symplectic form, we show that there exist canonically conjugate, orthogonal, metric submanifolds if and only if the original gauged space is Euclidean or signature 0. In the Euclidean cases, the resultant configuration space must be Lorentzian. Therefore, in this context, time may be viewed as a derived property of general relativity.

Conformal, concircular, quasi-conformal and conharmonic curvature tensors play an important role in Riemannian geometry. In this paper, we study on normal complex contact metric manifolds under flatness conditions of these tensors.

The complete structure of the Casimir ${{𝒲}{𝒜}}_N$ algebras is shown to exist in such a way that the Casimir ${{𝒲}{𝒜}}_N$ algebra is a kind of truncated type of ${{𝒲}}_{\infty }$ algebra both in the primary and in the quadratic basis, first using the associativity conditions in the basis of primary fields and second using the Miura basis coming from the free field realization as a different basis. We can conclude that the Casimir ${{𝒲}{𝒜}}_N$ algebra is a kind of truncated type of ${{𝒲}}_{\infty }$ algebra, so it is clear from any construction of ${{𝒲}}_{\infty }$ algebra that by putting infinite number of fields $W_s$ with $s>N$ to zero, we arrive at the Casimir ${{𝒲}{𝒜}}_N$ algebra. We concentrated in this work only for the particular case of ${{𝒲}{𝒜}}_5$ algebra since this example gives us explicitly a method on how to deal with the general case $N$.

With the extended navigation data, we consider the generalized Zermelo navigation on Riemannian manifolds, admitting a space-dependent ship’s speed in the presence of perturbation determined by a weak velocity vector field, with application of Finsler metric of Randers type. The approach is shown via indicatrix and inner product. We also compare our findings in the context of conformality for the cases of weak and critical winds. The study is illustrated with the example in dimension 2.