arXiv Math @arxiv_math@qoto.org

In this article, we study twisted derivations of cyclic group rings. Let $R$ be a commutative ring with unity, $G$ be a finite cyclic group, and ($σ, τ$) be a pair of $R$-algebra endomorphisms of the group algebra $RG$, which are $R$-linear extensions of the group endomorphisms of $G$. In this article, we give two characterizations concerning $(σ, τ)$-derivations of the group ring $RG$. First, we develop a necessary and sufficient condition for a $(σ, τ)$-derivation of $RG$ to be inner. Second, we provide a necessary and sufficient condition for an $R$-linear map $D: RG \rightarrow RG$ with $D(1) = 0$ to be a $(σ, τ)$-derivation. We also illustrate our theorems with the help of examples. As a consequence of these two characterizations, we answer the well-known twisted derivation problem for $RG$: Under what conditions are all $(σ, τ)$-derivations of $RG$ inner? Or is the space of outer $(σ, τ)$-derivations trivial? More precisely, we give a sufficient condition under which all $(σ, τ)$-derivations of $RG$ are inner and a sufficient condition under which $RG$ has non-trivial outer $(σ, τ)$-derivations. Our result helps in generating several examples of non-trivial outer derivations.

This letter explores the implementation of a safe control law for systems of dynamically coupled cooperating agents. Under a CBF-based collaborative safety framework, we examine how the maximum safety capability for a given agent, which is computed using a collaborative safety condition, influences safety requests made to neighbors. We provide conditions under which neighbors may be resilient to non-compliance of neighbors to safety requests, and compute an upper bound for the total amount of non-compliance an agent is resilient to, given its 1-hop neighborhood state and knowledge of the network dynamics. We then illustrate our results via simulation on a networked susceptible-infected-susceptible (SIS) epidemic model.

Let $X$ be a double EPW sextic, and $ι$ its anti-symplectic involution. We relate the $ι$-anti-invariant part of the Chow group of zero-cycles of $X$ with Voisin's rational orbit filtration. For a general double EPW sextic $X$, we also relate the anti-invariant part of the Chow motive of $X$ with the motive of a Gushel-Mukai fourfold. As an application, we obtain a similar result for certain Fano varieties of lines in cubics with infinite-order birational automorphisms.

Wireless communications using light waves are called visible light communication. A technique called Physical-layer Network coding allows several users to communicate simultaneously over the same channel in a distributed space-time Coding scheme. Herein, we report a new PNC-VLC one-to-many scheme using physical-layer network coding in VLC system for enhancing throughput by boosting data transmission at relay nodes. We develop an integrated PNC-VLC framework including physical-layer signal processing algorithms and medium access control approaches. Our proposed technique is then modelled mathematically and compared with some of the VLC and PNC-VLC schemes under various channel conditions and scenarios. Our investigation revealed that our proposed technique performed better than traditional PNC schemes in achieving better Bit Error Rate performance in different channel

In this paper, the $C^*$-algebra of the seven-dimensional un-decomposable nilpotent Lie group is characterized explicitly for the first time(see \cite{chin}). Furthermore, the topology of its spectrum is described as a preparation for the analysis of its $C^*$-algebra. Then, the operator-valued Fourier transform is employed to translate the given C*-algebra into the algebra of bounded operator fields through its spectrum. We find the conditions satisfied by the image of the goal to characterize them by these conditions, namely the "norm controlled dual limit" (NCDL)-conditions (see \cite{Lud-Reg}). The methods used for the nilpotent Lie groups are different for each. We consider the co-adjoint orbits and for our case, we use the Kirillov theory.

For $A \in M_m(\mathbb{C})$, Yamamoto's generalization of the spectral radius formula (1967) asserts that $\lim_{n \to \infty} s_j(A^n)^{\frac{1}{n}}$ is equal to the $j^{\textrm{th}}$-largest eigenvalue-modulus of $A$, where $s_j (A^n)$ denotes the $j^{\textrm{th}}$-largest singular value of $A^n$. Let $\mathscr{H}$ be a complex Hilbert space, and for an operator $T \in \mathscr{B}(\mathscr{H})$, let $|T| := (T^*T)^{\frac{1}{2}}$. For $A$ in $\mathscr{B}(\mathscr{H})$, we refer to the sequence, $\{ |A^n|^{\frac{1}{n}} \}_{n \in \mathbb{N} }$, as the $\textit{normalized power sequence}$ of $A$. As our main result, we prove a spatial form of Yamamoto's theorem which asserts that the normalized power sequence of a spectral operator in $\mathscr{B}(\mathscr{H})$ converges in norm, and provide an explicit description of the limit in terms of its idempotent-valued spectral resolution. Our approach substantially generalizes the corresponding result by the first-named author in the case of matrices in $M_m(\mathbb{C})$, and supplements the Haagerup-Schultz theorem on SOT-convergence of the normalized power sequence of an operator in a $II_1$ factor.

We show how to freely add the first layer of quantifier alternation depth to a Boolean doctrine over a small base category. This amounts to a doctrinal version of Herbrand's theorem for formulas with quantifier alternation depth at most one modulo a universal theory. To achieve this, we characterize, within the doctrinal setting, the classes $A$ of quantifier-free formulas for which there is a model $M$ such that $A$ is precisely the class of formulas whose universal closure is valid in $M$.

One of the most important sequences in enumerative combinatorics is OEIS sequence A85, the number of involutions of length n. In the Art of Computer Programming, vol. 3, Don Knuth derived the O(1/n) asymptotic formula for these numbers. In this modest tribute to our two heroes, Neil Sloane who just turned 85, and Don Knuth who was 85 a year ago, we go all the way to an $O(1/n^{85})$ asymptotic formula.

Randomized arcade processes are a class of continuous stochastic processes that interpolate in a strong sense, i.e., omega by omega, between any given ordered set of random variables, at fixed pre-specified times. Utilizing these processes as generators of partial information, a class of continuous-time martingale -- the filtered arcade martingales (FAMs) -- is constructed. FAMs interpolate through a sequence of target random variables, which form a discrete-time martingale. The research presented in this paper treats the problem of selecting the worst martingale coupling for given, convexly ordered, probability measures contingent on the paths of FAMs that are constructed using the martingale coupling. This optimization problem, that we term the information-based martingale optimal transport problem (IB-MOT), can be viewed from different perspectives. It can be understood as a model-free construction of FAMs, in the case where the coupling is not determined, a priori. It can also be considered from the vantage point of optimal transport (OT), where the problem is concerned with introducing a noise factor in martingale optimal transport, similarly to how the entropic regularization of optimal transport introduces noise in OT. The IB-MOT problem is static in its nature, since it is concerned with finding a coupling, but a corresponding dynamical solution can be found by considering the FAM constructed with the identified optimal coupling. Existence and uniqueness of its solution are shown, and an algorithm for empirical measures is proposed.

The $σ_k(A_g)$ curvature and the boundary $\mathcal{B}^k_g$ curvature arise naturally from the Chern--Gauss--Bonnet formula for manifolds with boundary. In this paper, we prove a Liouville theorem for the equation $σ_k(A_g)=1$ in $\overline{\mathbb{R}^n_+}$ with the boundary condition $\mathcal{B}^k_g=c$ on $\partial\mathbb{R}^n_+$, where $g=e^{2v}|dx|^2$ and $c$ is some nonnegative constant. This extends an earlier result of Wei, which assumes the existence of $\lim_{|x|\to\infty}(v(x)+2\log|x|)$. In addition, we establish a local gradient estimate for solutions of such equations, assuming an upper bound on the solution $v$.

We find a Floer theoretic approach to obtain the transpose polynomial $W^T$ of an invertible curve singularity $W$. This gives an intrinsic construction of the mirror transpose polynomial and enables us to define a canonical $A_\infty$-functor that takes Lagrangians in the Milnor fiber of W and converts them into matrix factorizations of $W^T$. We find Lagrangians in the Milnor fiber of $W$ that are mirror to the indecomposable matrix factorizations of $W^T$ when $W^T$ is ADE singularity and discover that Auslander-Reiten exact sequences can be realized as surgery exact triangles of Lagrangians in the mirror. There are two primary steps in the Floer theoretic method for obtaining a transposition polynomial: To get a Lagrangian $L$ and corresponding disc potential function $W_L$, we first determine the quotient $X$ by the maximal symmetry group for the Milnor fiber. Second, we define a class $Γ$ of symplectic cohomology of $X$ based on the monodromy of the singularity $W$. Another disc counting function, $g$, is defined by the closed-open image of $Γ$ on $L$. We demonstrate that restricting to the hypersurface $g = 0$ transforms the disc potential function $W_L$ into the transpose polynomial W T. This second step is the mirror of taking the cone of quantum cap action by the monodromy class $Γ$.

The theory of motivic superpolynomials is developed, including its extension to algebraic links, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis and DAHA recurrences for iterated torus links. The DAHA construction is provided. Applications are considered to affine Springer fibers and compactified Jacobians of type $A_n$ in the most general case (for arbitrary characteristic polynomials), rho-invariants of algebraic knots, motivic formulas for the DAHA-Jones polynomials, including the case of $A_1$. The key theme is the conjectural coincidence of motivic superpolynomials with the DAHA ones, which can be interpreted as a far-reaching generalization of the Shuffle Conjecture. The 2nd connection conjecture links the superpolynomials to the $L$-functions. DAHA theory is adjusted to the definition and main properties of superpolynomials. The motivic tools are systematically exposed, a certain unification of Schubert Calculus with the theory of plane curve singularities.

In this work, we study the composition operators on the little Lipschitz space ${\mathcal L}_0$ of a rooted tree $T$ defined as the subspace of the Lipschitz space ${\mathcal L}$, introduced in Colonna and Easley [4], that consists of the complex-valued functions $f$ on $T$ such that $$ \lim_{|v|\to\infty}|f(v)-f(v^-)|=0, $$ where $v^-$ is the vertex adjacent to the vertex $v$ in the path from the root to $v$ and $|v|$ denotes the number of edges from the root to $v$. Specifically, we give several sufficient conditions on a Lipschitz self-map $φ$ on $T$ for which the composition operator $C_φ$ maps ${\mathcal L}_0$ into itself, and estimate its operator norm. In addition we study the spectrum of $C_φ$ and the hypercyclicity of a constant multiple of $C_φ$.

A previous work introduced pair space, which is spanned by the center of mass of a system and the relative positions (pair positions) of its constituent bodies. Here, I show that in the $N$-body Newtonian problem, a configuration that does not remain on a fixed line in space is a central configuration if and only if it conserves all pair angular momenta. For collinear systems, I obtain a set of equations for the ratios of the relative distances of the bodies, from which I derive some bounds on the minimal length of the line. For the non-collinear case I derive some geometrical relations, independent of the masses of the bodies. These are necessary conditions for a non-collinear configuration to be central. They generalize, to arbitrary $N$, a consequence of the Dziobek relation, which holds for $N=4$.

I introduce an extended configuration space for classical mechanical systems, called pair-space, which is spanned by the relative positions of all the pairs of bodies. To overcome the non-independence of this basis, one adds to the Lagrangian a term containing auxiliary variables. As a proof of concept, I apply this representation to the three-body problem with a generalized potential that depends on the distance $r$ between the bodies as $r^{-n}$. I obtain the equilateral and collinear solutions (corresponding to the Lagrange and Euler solutions if $n=1$) in a particularly simple way. In the collinear solution, this representation leads to several new bounds on the relative distances of the bodies.

Let $M, N$ be free modules over a Noetherian commutative ring $R$ and let $F$ be a field whose cardinality does not exceed the continuum. We prove the following: 1) The Injective Continuum Function Hypothesis, ICF, is equivalent to assertion that [ any two F-vector spaces are isomorphic iff their duals are so.] 2) If the dual of $M$ is a projective $R$-module and rank$_R(M)$ is infinite then the ring $R$ is Artinian. 3) If $R$ is Artinian and card$(R)$ does not exceed the continuum then the dual of $M$ is free. 4) If $M, N$ have isomorphic duals then they are themselves isomorphic (over $R$), when rank$_R(M)$ is not an $ω$-measurable cardinal and $R$ is a non-Artinian ring that is either Hilbert or countable. 5) If $R$ is a non-local domain then $R$ is a half-slender ring. We also prove that if the powersets of two given sets have equal cardinalities then there is a bijection from the one powerset to the other that preserves the symmetric difference of sets.

Cosmological correlators encode statistical properties of the initial conditions of our universe. Mathematically, they can often be written as Mellin integrals of a certain rational function associated to graphs, namely the flat space wavefunction. The singularities of these cosmological integrals are parameterized by binary hyperplane arrangements. Using different algebraic tools, we shed light on the differential and difference equations satisfied by these integrals. Moreover, we study a multivariate version of partial fractioning of the flat space wavefunction, and propose a graph-based algorithm to compute this decomposition.

Let $\mathcal{D}$ be the whole space $\mathbb{R}^d$ or a sphere of radius $R>0$ in $\mathbb{R}^d$ with $1\leq d\leq3$. Consider the following cubic-quintic energy functional \begin{gather*} \begin{aligned} E_{N}(φ)=\frac{1}{2}\int_{\mathcal{D}}|\nabla φ|^2\,dx-\frac{N}{4}\int_{ \mathcal{D}}|φ|^{4}\,dx+\frac{N^{2}}{6}\int_{\mathcal{D}}|φ|^6\,dx \end{aligned} \end{gather*} minimized in $\left\lbrace φ\in H^1_0(\mathcal{D}): \int_{\mathcal{D}}|φ|^2\,dx=1\right\rbrace$. Firstly, we prove the limiting behavior of the ground state $φ_N$ as $N\to+\infty$, which corresponds to the \emph{Thomas-Fermi limit}. The limit profile is given by the Thomas-Fermi minimizer $u^{TF}=\sqrt{{3\cdot\mathbb{1}_Ω}/{4}}(x)$ with $Ω=B(0,\sqrt[d]{4/(3ω_d)})$. Moreover, we obtain a sharp vanishing rate for $φ_N$ that $\|φ_{N}\|_{L^\infty(\mathcal{D})}\sim N^{-{1}/{2}}$ as $N\to+\infty$. Finally, we prove that $\left\|N^{{1}/{2}}φ_{N}\left(N^{{1}/{d}}x\right)-u^{TF}(x)\right\|_{L^\infty\left(Ω\right)}\lesssim N^{-1/(2d)}$ as $N\to+\infty$.

Forward-backwards stochastic differential equations (FBSDEs) are central in optimal control, game theory, economics, and mathematical finance. Unfortunately, the available FBSDE solvers operate on \textit{individual} FBSDEs, meaning that they cannot provide a computationally feasible strategy for solving large families of FBSDEs as these solvers must be re-run several times. \textit{Neural operators} (NOs) offer an alternative approach for \textit{simultaneously solving} large families of FBSDEs by directly approximating the solution operator mapping \textit{inputs:} terminal conditions and dynamics of the backwards process to \textit{outputs:} solutions to the associated FBSDE. Though universal approximation theorems (UATs) guarantee the existence of such NOs, these NOs are unrealistically large. We confirm that ``small'' NOs can uniformly approximate the solution operator to structured families of FBSDEs with random terminal time, uniformly on suitable compact sets determined by Sobolev norms, to any prescribed error $\varepsilon>0$ using a depth of $\mathcal{O}(\log(1/\varepsilon))$, a width of $\mathcal{O}(1)$, and a sub-linear rank; i.e. $\mathcal{O}(1/\varepsilon^r)$ for some $r<1$. This result is rooted in our second main contribution, which shows that convolutional NOs of similar depth, width, and rank can approximate the solution operator to a broad class of Elliptic PDEs. A key insight here is that the convolutional layers of our NO can efficiently encode the Green's function associated to the Elliptic PDEs linked to our FBSDEs. A byproduct of our analysis is the first theoretical justification for the benefit of lifting channels in NOs: they exponentially decelerate the growth rate of the NO's rank.

An overview of the authors' ideas about the process of completing a $p$-adically normed space in the setting of vertex operator algebras. We focus in particular on the $p$-adic Heisenberg VOA and its connections with $p$-adic modular forms.