相关论文: Second order arithmetic means in operator ideals
Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory,…
We study first order differential operators with constant coefficients. The main question is under what conditions a generalized Poincar\'e inequality holds. We show that the constant rank condition is sufficient. The concept of the…
We consider the vanishing ideal of an arrangement of linear subspaces in a vector space and investigate when this ideal can be generated by products of linear forms. We introduce a combinatorial construction (blocker duality) which yields…
A necessary and sufficient condition is provided for the solvability of a binomial congruence with a composite modulus, circumventing its prime factorization. This is a generalization of Euler's Criterion through that of Euler's Theorem,…
A semiring is uniserial if its ideals are totally ordered by inclusion. First, we show that a semiring $S$ is uniserial if and only if the matrix semiring $M_n(S)$ is uniserial. As a generalization of valuation semirings, we also…
A notion of arithmetic similarity between number fields is defined by requiring equality of some arithmetic statistics over all but finitely many rational primes. The exceptional set is empty in all previously studied cases, but existing…
We consider a class of linear ODEs of second order with variable coefficients and construct its Lie algebra of Lie group of equivalence transformations. Further we find invariants and differential invariants of this Lie algebra and by using…
For a general class of non-negative functions defined on integral ideals of number fields, upper bounds are established for their average over the values of certain principal ideals that are associated to irreducible binary forms with…
Let $R$ be a commutative ring with unity $(1\not=0)$ and let $\mathfrak{J}(R)$ be the set of all ideals of $R$. Let $\phi:\mathfrak{J}(R)\rightarrow\mathfrak{J}(R)\cup\{\emptyset\}$ be a reduction function of ideals of $R$ and let…
The consistency of a second-order version of a theorem of Morley on the number of countable models was proved in arXiv:2107.07636 with the aid of large cardinals. We here dispense with them.
We study the reverse mathematics of interval orders. We establish the logical strength of the implications between various definitions of the notion of interval order. We also consider the strength of different versions of the…
Dual numbers and their higher order version are important tools for numerical computations, and in particular for finite difference calculus. Based upon the relevant algebraic rules and matrix realizations of dual numbers, we will present a…
In this paper we study the existence of continuous solutions and their constructions for a second order iterative functional equation, which involves iterate of the unknown function and a nonlinear term. Imposing Lipschitz conditions to…
The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. As a `degenerate version' of this, we consider the normal ordering of a…
For a nonlinear ordinary differential equation with time delay, the differentiation of the solution with respect to the delay is investigated. Special emphasis is laid on the second-order derivative. The results are applied to an associated…
We introduce a total order and the absolute value function for dual numbers. The absolute value function of dual numbers are with dual number values, and have properties similar to the properties of the absolute value function of real…
Consider a grade 2 perfect ideal $I$ in $R=k[x_1,\cdots,x_d]$ which is generated by forms of the same degree. Assume that the presentation matrix $\varphi$ is almost linear, that is, all but the last column of $\varphi$ consist of entries…
This paper presents an up-to-date and refined version of the SCL calculus for first-order logic without equality. The refinement mainly consists of the following two parts: First, we incorporate a stronger notion of regularity into…
Let $E$ be a module of projective dimension one over $R=k[x_1,\ldots,x_d]$. If $E$ is presented by a matrix $\varphi$ with linear entries and the number of generators of $E$ is bounded locally up to codimension $d-1$, the Rees ring…
We show that H\"older continuity of the gradient is not only a sufficient condition, but also a necessary condition for the existence of a global upper bound on the error of the first-order Taylor approximation. We also relate this global…