Ideal Theorem Research Articles

Reverse lex ideals

We study reverse lex ideals in a polynomial ring, and compare their properties to those of lex ideals. In particular we provide an analogue of Green's Theorem for reverse lex ideals. We also compare the Betti numbers of strongly stable and square-free strongly stable monomial ideals to those of reverse lex ideals.

Some odd permutations

I recently stumbled on an identity (Theorem 2) that related two types of permutations. It looked like one of those serendipitous pieces of mathematics that led to a simple enumerative result; mostly when this happens it leads joyously to an enumerative proof that was ‘blindingly’ obvious. Except in this case, it wasn’t, I don’t think.

Combinatorial aspects of free associative algebras and cohomologies of Lie p-algebras with one defining relation

Let R and Q be elements of a free, associative, finitely generated algebra A‹X› over a field k. Assume that a leading homogeneous part Q v does not have two-sided divisors and RQ = QR and R v = Q t v . In this paper, the solutions of the equation Σ i x i Ry i = z in A‹X› are found and with their help, the identity theorem and Freiheitssatz for a finitely generated associative algebra k‹X; R = 0›with one defining relation are proved. As a consequence, similar theorems for Lie p-algebras with one defining relation are proved; these results are applied to the proof of the periodicity of cohomologies of Lie p-algebras with one defining relation.

Identity theorem for bounded p-adic meromorphic functions

Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences ( a n ) n ∈ N in a disk d ( 0 , R − ) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f ( a n ) = 0 ∀ n ∈ N implies f = 0 ), identity sequences (when lim n → + ∞ f ( a n ) = 0 implies f = 0 ) and analytic boundaries (when lim sup n → ∞ | f ( a n ) | = ‖ f ‖ ). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions.

Actions of Groups of Finite Morley Rank on Small Abelian Groups

AbstractWe classify actions of groups of finite Morley rank on abelian groups of Morley rank 2: there are essentially two, namely the natural actions ofSL(V)andGL(V)withVa vector space of dimension 2. We also prove an identification theorem for the natural module of SL2in the finite Morley rank category.

Uniqueness and structure of solutions to the Dirichlet problem for an elliptic system

In this paper, we consider the Dirichlet problem for an elliptic system on a ball in R 2 . By investigating the properties for the corresponding linearized equations of solutions, and adopting the Pohozaev identity and Implicit Function Theorem, we show the uniqueness and the structure of solutions.

Function S-rough sets and law identification

By introducing element equivalence class that proposes dynamic characteristic into Pawlak Z rough sets theory, the first author of this paper improved Pawlak Z rough sets and put forward S-rough sets (singular rough sets). S-rough sets are defined by element equivalence class that proposes dynamic characteristic. S-rough sets have dynamic characteristic. By introducing the function equivalence class (law equivalence class) that proposes dynamic characteristic into S-rough sets, the first author improved S-rough sets and put forward function S-rough sets (function singular rough sets). Function S-rough sets have dynamic characteristic and law characteristic, and a function is a law. By using function S-rough sets, this paper presents law identification, law identification theorem, and law identification criterion and applications. Function S-rough sets are a new research direction of rough sets theory, and it is also a new tool to the research of system law identification.

Finite groups with minimal 1-PIM

Let \(\mathbb F\) be a field of characteristic \(\ell > 0\) and let G be a finite group. It is well-known that the dimension of the minimal projective cover \(\Phi_1^G\) (the so-called 1-PIM) of the trivial left \(\mathbb F[G]\) -module is a multiple of the \(\ell\) -part \(|G|_\ell\) of the order of G. In this note we study finite groups G satisfying \(\dim_{\mathbb F}(\Phi_1^G)=|G|_\ell\) . In particular, we classify the non-abelian finite simple groups G and primes \(\ell\) satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers \(\ell\) (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number \(\ell\in\{2,3,5\}\) implies the existence of an \(\ell^\prime\) -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Proposition 2.2).

An estimation problem for the shape of a domain varying with time via parabolic equations

This paper treats the inverse problem determining the shape of a multi-dimensional domain varying with time from measurements of the temperature on an accessible portion of the boundary. On the shape with only Lipschitz continuity on the unknown portion of the boundary, we provide the unique identification theorem and a reconstruction procedure based on a suitably linearized problem, which is formulated by an initial-boundary value problem for a parabolic equation with a mixed boundary condition. For the reconstruction procedure, its convergence and stability are verified, and further its effectiveness is examined in numerical examples.

An ideal topology type convergent theorem on scale effect algebras

The famous Antosik-Mikusinski convergent theorem on the Abel topological groups has very extensive applications in measure theory, summation theory and other analysis fields. In this paper, we establish the theorem on a class of effect algebras equipped with the ideal topology. This paper shows also that the ideal topology of effect algebras is a useful topology in studying the quantum logic theory.

Group-theoretic compactification of Bruhat–Tits buildings

Let G F denote the rational points of a semisimple group G over a non-archimedean local field F, with Bruhat–Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of G F in the Chabauty topology, which enables us to compactify the vertices of X. We obtain a structure theorem showing that the Bruhat–Tits buildings of the Levi factors all lie in the boundary of the compactification. Then we obtain an identification theorem with the polyhedral compactification (previously defined in analogy with the case of symmetric spaces). We finally prove two parametrization theorems extending the Bruhat–Tits dictionary between maximal compact subgroups and vertices of X: one is about Zariski connected amenable subgroups and the other is about subgroups with distal adjoint action.

On the Nonparametric Identification of Nonlinear Simultaneous Equations Models: Comment on Brown (1983) and Roehrig (1988)

This note revisits the identification theorems of Brown (1983) and Roehrig (1988). We describe an error in the proofs of the main identification theorems in these papers, and provide an important counterexample to the theorems on the identification of the reduced form. Specifically, the reduced form of a nonseparable simultaneous equations model is not identified even under the assumptions of these papers. We provide conditions under which the reduced form is identified and is recoverable using the distribution of the endogenous variables conditional on the exogenous variables. However, these conditions place substantial limitations on the structural model. We conclude the note with a conjecture that it may be possible to use classical exclusion restrictions to recover some of the key implications of the theorems in more general settings.

Generalized Cochrane sums and Cochrane–Hardy sums

In this paper, the generalized Cochrane sums and Cochrane–Hardy sums are defined. The arithmetic properties of the generalized Cochrane sums are studied, and the Cochrane–Hardy sums are expressed in terms of the generalized Cochrane sums. Analogues of Subrahmanyam's identity and Knopp's theorem are given and proved. Finally, the hybrid mean value of generalized Cochrane sums, Cochrane–Hardy sums and Kloosterman sums is studied, and a few asymptotic formulae are obtained.

AI and Science's Lost Realm

We demand phenomenality for scientific evidence, are utterly dependent on it for scientific evidence (it's all there is), and are unable to observe it. Phenomenality appears beyond empirical science's reach. A modern form of this decree is the mind-brain identity theorem (MBIT). Space limitations prevent a detailed discussion, but a large and complex history has embedded the MBIT at the heart of current empirical science of the mind

On the Nonparametric Identification of Nonlinear Simultaneous Equations Models: Comment on B. Brown (1983) and Roehrig (1988)

This note revisits the identification theorems of B. Brown (1983) and Roehrig (1988). We describe an error in the proofs of the main identification theorems in these papers, and provide an important counterexample to the theorems on the identification of the reduced form. Specifically, the reduced form of a nonseparable simultaneous equations model is not identified even under the assumptions of these papers. We provide conditions under which the reduced form is identified and is recoverable using the distribution of the endogenous variables conditional on the exogenous variables. However, these conditions place substantial limitations on the structural model. We conclude the note with a conjecture that it may be possible to use classical exclusion restrictions to recover some of the key implications of the theorems in more general settings.

Nonequilibrium Steady State of a Nanometric Biochemical System: Determining the Thermodynamic Driving Force from Single Enzyme Turnover Time Traces

A single enzyme molecule in a living cell is a nanometric system that catalyzes biochemical reactions in a nonequilibrium steady-state condition. The chemical driving force, Deltamu, is an important thermodynamic quantity that determines the extent to which the reaction system is away from equilibrium. Here we show that Deltamu for an enzymatic reaction in situ can be determined from the nonequilibrium time traces for enzymatic turnovers of individual enzyme molecules, which can now be recorded experimentally by single-molecule techniques. Three different Deltamu estimators are presented from principles of nonequilibrium statistical mechanics: fluctuation theorem, Kawasaki identity, and fluctuation dissipation theorem, respectively. In particular, a maximum likelihood estimation method of Deltamu has been derived based on fluctuation theorem. The statistical precisions of these three Deltamu estimators are analyzed and compared for experimental time traces with finite lengths.

Factor Rings and Splitting Ideals of Self-Injective, CS and Baer Rings#

ABSTRACT We study conditions on an ideal A of a self-injective R such that the factor ring R/ A is again self-injective, extending certain of our results for PF rings (Faith, 2006). We also consider the same question for p -injective, and for CS -rings. For the CS -rings we consider conditions under which A splits off as a ring direct factor, equivalently, when A is generated by a central idempotent. Definitive results are obtained for an ideal A which is semiprime as a ring, that is, has no nilpotent ideals except zero, and which is a right annihilator ideal. Then A is said to be an r -semiprime right annulet ideal, and is generated by a central idempotent in the following cases: (1) whenever A is generated by an idempotent as a right (or left) ideal (Theorems 3.4, 3.6); (2) in any Baer ring R (Theorem 3.5); (3) in any right and left CS -ring R (Theorem 4.2), and (4) in any right nonsingular right CS -ring R (Theorem 5.5). These results also generalize results of the author in Faith (1985), where it is proven that the maximal regular ideal M( R) splits off in any right and left continuous ring. The results are applied in Section 6 to extend theorems of Faith (1996) characterizing VNR rings, and, as the title of Faith (1996) suggests, extend the conjecture of Shamsuddin.

The Structure Theorem for Complete Intersections of Grade 4

Serre showed that a Gorenstein ideal of grade 2 is a complete intersection, and Buchsbaum and Eisenbud proved a structure theorem for Gorenstein ideals of grade 3. It is found that a certain complete matrix defines a perfect ideal [Formula: see text]. As an application, we present a structure theorem for complete intersections of grade 4.

Explorations of the extended ncKP hierarchy

A recently obtained extension (xncKP) of the Moyal-deformed KP hierarchy (ncKP hierarchy) by a set of evolution equations in the Moyal-deformation parameters is further explored. Formulae are derived to compute these equations efficiently. Reductions of the xncKP hierarchy are treated, in particular to the extended ncKdV and ncBoussinesq hierarchies. Furthermore, a good part of the Sato formalism for the KP hierarchy is carried over to the generalized framework. In particular, the well-known bilinear identity theorem for the KP hierarchy, expressed in terms of the (formal) Baker-Akhiezer function, extends to the xncKP hierarchy. Moreover, it is demonstrated that N-soliton solutions of the ncKP equation are also solutions of the first few deformation equations. This is shown to be related to the existence of certain families of algebraic identities.

Hitchin's Connection and Differential Operators with Values in the Determinant Bundle

For a family of smooth curves, we have the associated family of moduli spaces of stable bundles with fixed determinant on the curves. There exists a so called theta line bundle on the family of moduli spaces. When the Kodaira–Spencer map of the family of curves is an isomorphism, we prove in this paper an identification theorem between sheaves of differential operators on the theta line bundle and higher direct images of vector bundles on curves. As an application, the so called Hitchin's connection on the direct image of (powers of) theta line bundle is derived naturally from the identification theorem. A logarithmic extension of Hitchin's connection to certain singular stable curves is also presented in this paper.