Quadratic nonsymmetric quaternary operads

We use computational linear algebra and commutative algebra to study spaces of relations satisfied by quadrilinear operations. The relations are analogues of associativity in the sense that they are quadratic (every term involves two operations) and nonsymmetric (every term involves the identity permutation of the arguments). We focus on determining those quadratic relations whose cubic consequences have minimal or maximal rank. We approach these problems from the point of view of the theory of algebraic operads.     

Cyclic terms for SD⋁ varieties revisited

We present a direct proof showing that every finite algebra generating a congruence join semidistributive variety has a cyclic term.     

Integral powers of the Sturm-Liouville operator and the theory of partitions

We use unrestricted partitions and a recurrence relation to specify every term of every integral power of the Sturm-Liouville (or one-dimensional Schrödinger) operator. Our purpose is to exhibit techniques that might yield proofs of our combinatorial conjectures on the conservation laws of the Korteweg-de Vries equation. At the end some open problems are stated.     

On the Liu–Floudas Convexification of Smooth Programs

It is well known that a twice continuously differentiable function can be convexified by a simple quadratic term. Here we show that the convexification is possible also for every Lipschitz continuously differentiable function. This implies that the Liu–Floudas convexification works for, loosely speaking, almost every smooth program occurring in practice. Research supported by an NSERC of Canada grant     

UPWIND SPLITTING SCHEME FOR CONVECTION-DIFFUSION EQUATIONS

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Well Ordering Groups with no Monotone Arithmetic Progressions

Károlyi–Kós and Ardal–Brown–Jungic proved that every vector space over \(\mathbb {Q}\) has an ordering with no monotone three term arithmetic progression (3-AP). We show that every solvable group has a well ordering with no monotone 6-AP, and each hypoabelian group has an ordering omitting monotone 5-APs. Finally, we prove that every group has a well ordering with no infinite monotone AP.     

More Primes and Polynomials

Triplets of polynomials in four variables with rational coefficients and every term of arbitrarily high degree are exhibited. At least one polynomial in each triplet vanishes on infinitely many integer points whose coordinates are all prime.     

More on oscillation ofnth-order equations

In this paper we prove that a higher-order differential equation with one middle term has every bounded solution oscillatory. Moreover, the behavior of unbounded solutions is given. Two other results dealing with positive solutions are also given.     

Dualizable algebras with parallelogram terms

We prove that if A is a finite algebra with a parallelogram term that satisfies the split centralizer condition, then A is dualizable. This yields yet another proof of the dualizability of any finite algebra with a near unanimity term, but more importantly proves that every finite module, group or ring in a residually small variety is dualizable.     

A lexicographic path order with slow growing derivation bounds

This paper is concerned with implicit computational complexity of the exptime computable functions. Modifying the lexicographic path order, we introduce a path order EPO. It is shown that a termination proof for a term rewriting system via EPO implies an exponential bound on the lengths of derivations. The path order EPO is designed so that every exptime function is representable as a term rewrite system compatible with EPO (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gronwall-Bellman 型非线性差分不等式及在差分方程中的应用

In this paper,we establish some general sums-difference inequalities with two variables.The inequalities involve finite sum and every term contains the unknown function of the composite function with the power of pi.In the end,we study boundedness of the solution of the difference equations as applications.     

Finite degree: algebras in general and semigroups in particular

An algebra A has finite degree if its term functions are determined by some finite set of finitary relations on A. We study this concept for finite algebras in general and for finite semigroups in particular. For example, we show that every finite nilpotent semigroup has finite degree (more generally, every finite algebra with bounded p _n -sequence), and every finite commutative semigroup has finite degree. We give an example of a five-element unary semigroup that has infinite degree. We also give examples to show that finite degree is not preserved in general under taking subalgebras, homomorphic images, direct products or subdirect factors.     

A note on CIP varieties

It is proved that every variety satisfying the Congruence Intersection Property (CIP) is Abelian. In addition, a CM Abelian variety has the CIP if and only if it has a constant term operation. Finally, a CM variety is Abelian if and only if it has the weak CIP. Received October 8, 1998; accepted in final form January 5, 1999.     

A decidable theory of type assignment

This article investigates a theory of type assignment (assigning types to lambda terms) called ETA which is intermediate in strength between the simple theory of type assignment and strong polymorphic theories like Girard’s F (Proofs and types. Cambridge University Press, Cambridge, 1989). It is like the simple theory and unlike F in that the typability and type-checking problems are solvable with respect to ETA. This is proved in the article along with three other main results: (1) all primitive recursive functionals of finite type are representable in ETA; (2) every term typable in ETA has a unique normal form; (3) there is a function defined by ${{\varepsilon}_0}$ -recursion which takes every typable term to a natural number which is an upper bound to the lengths of all βη-reduction sequences starting with that term.     

An unsaturated numerical method for the exterior axisymmetric Neumann problem for Laplace’s equation

Basing on the fundamental ideas of Babenko, we construct a fundamentally new, unsaturated, numerical method for solving the axially symmetric exterior Neumann problem for Laplace’s equation. The distinctive feature of this method is the absence of the principal error term enabling us to automatically adjust to every class of smoothness of solutions natural in the problem.     

A fifth-order family of modified Newton methods

Addition of a correction term every other Newton iteration provides a fifth-order method for finding simple zeros of nonlinear functions. A two-parameter family of such methods is developed. Each family member requires the given function and its derivative to be evaluated at two points per step.Work supported by the British Science Research Council at the University of Dundee, and by the U.S. Atomic Energy Commission.     

Star-linear equational theories of groupoids

We prove that there are precisely six equational theories E of groupoids with the property that every term is E-equivalent to a unique linear term. Presented by J. Berman. Received November 11, 2004; accepted in final form March 12, 2006. The first and third authors were supported by the Ministry of Science and Environment of Serbia, grant no. 144011; the second and fifth authors were supported by MŠMT, research project MSM 0021620839, and by the Grant Agency of the Czech Republic, grant #201/02/0594; the fourth author was supported by the NSF grant #DMS-0245622.     

Covering in the lattice of equational theories and some properties of term finite theories

For countable languages, we completely describe those cardinals κ such that there is an equational theory which covers exactly κ other equational theories. For this task understanding term finite theories is helpful. A theoryT isterm finite provided {ψ:Tτϕ≈ψ} is finite for all terms ϕ. We develop here some fundamental properties of term finite theories and use them, together with Ramsey's Theorem, to prove that any finitely based term finite theory covers only finitely many others. We also show that every term finite theory possesses an independent base and that there are such theories whose pairwise joins are not term finite. The paper was written with the support of NSF Grant MCS-80-01778. Presented by B. Jónsson. Received July 22, 1980. Accepted for publication in final form March 19, 1981.     

Equational partiality

A special kind of partiality of heterogeneous algebraic structures is introduced. Every operator of a heterogeneous operator domain is associated with a set of term equations as necessary and sufficient domain condition.It is shown that some kind of hierarchy condition for the system of domain equations is equivalent to the condition that every injective weak homomorphism is a strong homomorphism which is equivalent to the statement that every bijective weak homomorphism is an isomorphism.On the base of this result the notions of a quasi-variety and of a variety of equationally partial heterogeneous algebras are suggested. The class of all small categories becomes a standard example of a variety of equational partial heterogeneous algebras.Presented by V. Trnková.