Product Theorem Research Articles

Entropy production and fluctuation theorems under feedback control: the molecularrefrigerator model revisited

We revisit the model of a Brownian particle in a heat bath submitted to an activelycontrolled force proportional to the velocity that leads to thermal noise reduction (colddamping). We investigate the influence of the continuous feedback on the fluctuations ofthe total entropy production and show that the explicit expression of the detailedfluctuation theorem involves different dynamics and observables in the forward andbackward processes. As an illustration, we study the analytically solvable case of aharmonic oscillator and calculate the characteristic function of the entropy production in anonequilibrium steady state. We then determine the corresponding large deviationfunction which results from an unusual interplay between ‘boundary’ and ‘bulk’contributions.

New Inversion Techniques for Some Integral Transforms via the Generalized Product Theorem of the Mellin Transform

We introduce the generalized product theorem for the Mellin transform, and we solve certain classes of singular integral equations with kernels coincided with conditions of this theorem. Also, new inversion techniques for the Wright, Mittag-Leffler, Stieltjes, and Widder potential transforms are obtained.

Rolling With Simplified Stream Function Velocity and Strain Rate Vector Inner Product

The dual-stream function velocity field is reduced in order to analyze two-dimensional plate broadside rolling in roughing. The strain rate vector inner product and integral mean value theorem, as well as co-line vector inner product are used respectively in plastic deformation power, friction losses and shear power. A theoretical solution of roll torque and separating force for the rolling is obtained and the calculated results by the solution are compared with those measured in broadside rolling on-line. It shows that both the force and torque calculated are higher than those of measured, but the maximum relative error between them is no more than 11%

On the duality theorem on an analytic variety

The duality theorem for Coleff-Herrera products on a complex manifold says that if $f = (f_1,\dots,f_p)$ defines a complete intersection, then the annihilator of the Coleff-Herrera product $\mu^f$ equals (locally) the ideal generated by $f$. This does not hold unrestrictedly on an analytic variety $Z$. We give necessary, and in many cases sufficient conditions for when the duality theorem holds. These conditions are related to how the zero set of $f$ intersects certain singularity subvarieties of the sheaf $\mathcal{O}_Z$.

Fluctuation theorems in the presence of information gain and feedback

In this study, we rederive the fluctuation theorems in the presence of feedback by assuming the known Jarzynski equality and detailed fluctuation theorems. We find that both the classical and quantum systems can be analyzed using a similar treatment in terms of state space trajectories. We first briefly reproduce the already known work theorems for a classical system in order to show its equivalence with the quantum treatment. We then extend the treatment to arrive at new results, namely the generalizations of Seifert’s entropy production theorem and the Hatano–Sasa fluctuation theorem, in the presence of feedback. We have also derived the extended version of the Tasaki–Crooks fluctuation theorem for a quantum particle in the presence of multiple loop feedback. For deriving the extended quantum fluctuation theorems, we have considered open systems. No assumption is made on the nature of environment and the strength of system–bath coupling. However, it is assumed that the measurement process involves classical errors.

Differential Transform Method for Nonlinear Initial-Value Problems by Adomian Polynomials

In this paper, the differential transformation method is modified to be easily employed to solve wide kinds of nonlinear initial-value problems. In this approach, the nonlinear term is replaced by its Adomian polynomials for the index k, and hence the dependent variable components are replaced in the recurrence relation by their corresponding differential transform components of the same index. Thus the nonlinear initial-value problem can be easily solved with less computational effort. New theorems for product and integrals of nonlinear functions are introduced. In order to show the power and effectiveness of the present modified method and to illustrate the pertinent features of related theorems, several numerical examples with different types of nonlinearities are considered.

Central limit theorem for stationary products of toral automorphisms

Let $(A_n(\omega))$ be a stationary process in ${\mathcal M}_d^*(\mathbb{Z})$. For a Holder function $f$ on $\mathbb{T}^d$ we consider the sums $\sum_{k=1}^n f(^t\hskip -3pt A_k(\omega) \, ^t\hskip -3pt A_{k-1}(\omega) \cdots ^t\hskip -3pt A_1(\omega) \, x {\rm mod } 1)$ and prove a Central Limit Theorem for a.e. $\omega$ in different situations in particular for kicked stationary processes. We use the method of multiplicative systems of Komlos and the Multiplicative Ergodic Theorem.

Strong Direct Product Theorems for Quantum Communication and Query Complexity

A strong direct product theorem (SDPT) states that solving $n$ instances of a problem requires $\Omega(n)$ times the resources for a single instance, even to achieve success probability $2^{-\epsilon n}$ for a small enough constant $\epsilon>0.$ We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by the generalized discrepancy method, the strongest technique in that model. We prove that quantum query complexity obeys an SDPT whenever the query lower bound for a single instance is proved by the polynomial method, one of the two main techniques in that model. In both models, we prove the corresponding XOR lemmas and threshold direct product theorems.

Metabelian Lie powers of the natural module for a general linear group

Consider a free metabelian Lie algebra M of finite rank r over an infinite field K of prime characteristic p. Given a free generating set, M acquires a grading; its group of graded automorphisms is the general linear group G L r ( K ) , so each homogeneous component M d is a finite dimensional G L r ( K ) -module. The homogeneous component M 1 of degree 1 is the natural module, and the other M d are the metabelian Lie powers of the title. This paper investigates the submodule structure of the M d . In the main result, a composition series is constructed in each M d and the isomorphism types of the composition factors are identified both in terms of highest weights and in terms of Steinbergʼs twisted tensor product theorem; their dimensions are also given. It turns out that the composition factors are pairwise non-isomorphic, from which it follows that the submodule lattice is finite and distributive. By the Birkhoff representation theorem, any such lattice is explicitly recognizable from the poset of its join-irreducible elements. The poset relevant for M d is then determined by exploiting a 1975 paper of Yu.A. Bakhturin on identical relations in metabelian Lie algebras.

On two-part Sperner systems for regular posets (Extended Abstract)

Abstract In the Boolean lattice the BLYM inequality holds with equality if and only if the Sperner family consists of one complete level of subsets. In this paper we extend this strict BLYM-property to a subclass of normal posets. On the basis of this result we prove a strict two-part Sperner theorem of the direct product of any two posets from the same subclass.

Cut ideals of K4-minor free graphs are generated by quadrics

Cut ideals are used in algebraic statistics to study statistical models defined by graphs. Intuitively, topological restrictions on the graphs should imply structural statements about the corresponding cut ideals. Several theorems and many computer calculations support that. Sturmfels and Sullivant conjectured that the cut ideal is generated by quadrics if and only if the graph is free of K4-minors. Parts of the conjecture has been resolved by Brennan and Chen, and later by Nagel and Petrovic. We prove the full conjecture by introducing a new type of toric fiber product theorem.

Stratifying modular representations of finite groups

We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equiv- alently all) modules. One application is a proof of the telescope conjecture in this context. Others include new proofs of the tensor product theorem and of the classification of thick subcategories of the finitely generated modules which avoid the use of cyclic shifted subgroups. Along the way we establish similar classifications for differential graded modules over graded polynomial rings, and over graded exterior algebras.

The algebra of set functions I: The product theorem and duality

We give a comprehensive introduction to the algebra of set functions and its generating functions. This algebraic tool allows us to formulate and prove a product theorem for the enumeration of functions of many different kinds, in particular injective functions, surjective functions, matchings and colourings of the vertices of a hypergraph. Moreover, we develop a general duality theory for counting functions.

Simultaneous diophantine approximation in non-degenerate p-adic manifolds

An S-arithmetic Khintchine-type theorem for products of non-degenerate analytic p-adic manifolds is proved for the convergence case. In the p-adic case the divergence part is also obtained.

Stratifying triangulated categories

A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences which follow when T is stratified by R. Among them are a classification of the localizing subcategories of T in terms of subsets of the set of prime ideals in R; a classification of the thick subcategories of the subcategory of compact objects in T; and results concerning the support of the R-module of homomorphisms Hom_T^*(C,D) leading to an analogue of the tensor product theorem for support varieties of modular representation of groups.

ON THE LOCAL-INDICABILITY COHEN–LYNDON THEOREM

AbstractFor a group H and a subset X of H, we let HX denote the set {hxh−1 | h ∈ H, x ∈ X}, and when X is a free-generating set of H, we say that the set HX is a Whitehead subset of H. For a group F and an element r of F, we say that r is Cohen–Lyndon aspherical in F if F{r} is a Whitehead subset of the subgroup of F that is generated by F{r}. In 1963, Cohen and Lyndon (D. E. Cohen and R. C. Lyndon, Free bases for normal subgroups of free groups, Trans. Amer. Math. Soc. 108 (1963), 526–537) independently showed that in each free group each non-trivial element is Cohen–Lyndon aspherical. Their proof used the celebrated induction method devised by Magnus in 1930 to study one-relator groups. In 1987, Edjvet and Howie (M. Edjvet and J. Howie, A Cohen–Lyndon theorem for free products of locally indicable groups, J. Pure Appl. Algebra45 (1987), 41–44) showed that if A and B are locally indicable groups, then each cyclically reduced element of A*B that does not lie in A ∪ B is Cohen–Lyndon aspherical in A*B. Their proof used the original Cohen–Lyndon theorem. Using Bass–Serre theory, the original Cohen–Lyndon theorem and the Edjvet–Howie theorem, one can deduce the local-indicability Cohen–Lyndon theorem: if F is a locally indicable group and T is an F-tree with trivial edge stabilisers, then each element of F that fixes no vertex of T is Cohen–Lyndon aspherical in F. Conversely, by Bass–Serre theory, the original Cohen–Lyndon theorem and the Edjvet–Howie theorem are immediate consequences of the local-indicability Cohen–Lyndon theorem. In this paper we give a detailed review of a Bass–Serre theoretical form of Howie induction and arrange the arguments of Edjvet and Howie into a Howie-inductive proof of the local-indicability Cohen–Lyndon theorem that uses neither Magnus induction nor the original Cohen–Lyndon theorem. We conclude with a review of some standard applications of Cohen–Lyndon asphericity.

On wreath products of C-algebras, table algebras, and association schemes

In this paper we will first present characterizations of wreath products of C-algebras, and show that the dual of the wreath product of C-algebras is isomorphic to the wreath product of the duals of those C-algebras in the reverse order. Then we prove the Krull–Schmidt type theorems for the wreath products of table algebras as well as for the wreath products of association schemes.

Positional dimension-like functions of the type Ind

In Iliadis (2005) [4] positional dimension-like functions of the type ind are given. All these functions are studied only with respect to the property of universality. In a later paper by the present authors, and in two papers by V.V. Tkachuk (1981, 1982) (see [7,8]), these dimension-like functions have been studied with respect to the other standard properties of dimension theory. In R. Koga, Subspace-dimension with respect to total spaces, Master Thesis, Osaka Kyoiku University, 1998 (see also K.P. Hart, Jun-iti Nagata, J.E. Vaughan, Encyclopedia of General Topology, Elsevier Science Publishers, B.V., Amsterdam, 2004) a positional dimension-like function of the type Ind is given. Here we define new positional dimension-like functions of the type Ind, and present for all these functions, theorems concerning subspace theorems, partition theorems, sum theorems, and product theorems. Finally, we give some open questions concerning these functions.

The general Gaussian product theorem

The Gaussian Product Theorem between two 1s Gaussian Type Orbitals (GTOs) is extended to an arbitrary number of s-type functions, giving a compact formula which permits to express the condensed GTO result. Then, this new formulation is combined with the product of arbitrary GTO Cartesian angular parts, obtaining a finite expansion expressing this product as a multilinear combination of such functions sharing a common center. The combination of both results constitutes the General Gaussian Product Theorem (GGPT), a final compact expression for the product of an arbitrary number of Cartesian GTOs. The notation provides an easy way to express algebraically any general multicenter GTO product expression. It is also shown how by means of Nested Summation Symbols the computational implementation can be easily and elegantly achieved. Computational parallelizable schemes of the GGPT are provided.

Z-set unknotting in uncountable products of reals

We prove a version of Z-set unknotting theorem for uncountable products of real numbers.