1,721,002 research outputs found
On the minimality of the powers of the minimal omega-bounded abelian groups
No full textAn answer to the question whether the countably infinite power of a minimal countably compact abelian group is still minimal is given, using specific technique based on Pontryagin duality
Manipulation of Sums
No full textThis chapter deals with the calculus of finite sums: After examining some special techniques, we develop the general theory of finite calculus, the discrete analogue of differential calculus. The discrete primitives are the tool that enable to compute finite sums. We examine in detail the case of the sums of powers of consecutive natural numbers: quite surprisingly this leads to the Stirling numbers of second kind. A section is devoted to the inversion formula, a powerful tool in many mathematical fields: we use it here to obtain the discrete analogue of the Taylor expansion, an alternative short proof of both the number of derangements of a sequence and of surjective functions between two finite sets, and, finally, a more general version of the inclusion/exclusion principle
The Euler–Maclaurin formulas of order 1 and 2
No full textLet a< b be two integers and f: [ a, b] → R a function. In Chap. 6 we saw how to calculate the sum ∑a≤
Occupancy constraints
No full textThis chapter introduces a finer level of analysis for counting sequences or collections that are subject to some occupancy constraint, namely a constraint on the number of repetitions of its elements. Several problems are considered. As more unusual application in this framework, we prove the Leibniz rule for the derivatives of a product of functions, and count, in terms of the Catalan numbers, the Dyck sequences, i.e., the binary sequences of even length with equal number of 0’s and 1’s where, at each position, the number of 1’s does not exceed the number of 0’s
Reflexivity in derived categories
No full textAn adjoint pair of contravariant functors between abelian categories can be extended to the adjoint pair of their derived functors in the associated derived categories. We describe the reflexive complexes and interpret the achieved results in terms of objects of the initial abelian categories. In particular we prove that, for functors of any finite cohomological dimension, the objects of the initial abelian categories which are reflexive as stalk complexes form the largest class where a Cotilting Theorem in the sense of Colby and Fuller [CbF1, Ch. 5] works
Generating formal series and applications
No full textWe deepen here the insight on formal power series. We temporarily abandon formality and consider the notion of the convergence of a power series; we’ll see in particular how a smart choice of a closed form of a given power series is useful to recover the sum of the power series. A large part of the chapter is devoted to determining the generating formal series for some notable sequences, including sequences of binomial coefficients, harmonic numbers, Stirling and Bell numbers, Eulerian numbers, as well as sequences of integral powers. One section is devoted to the Bernoulli numbers: not only do they allow us to express the sum of consecutive m-th powers of the natural numbers (Faulhaber’s formula), but they turn out to be useful, as we shall see in Chap. 13, in approximating the sum of the consecutive values on the natural numbers of any given smooth function. Some useful estimates of the Bernoulli numbers are given via the Riemann zeta function, namely the sum of the series of the inverses of a given real power of the natural numbers. Finally, a section is devoted to the applications of formal power series to probabilities
Cauchy and Riemann sums, factorials, Ramanujan numbers and their approximations
No full textAfter a short recall of the basic asymptotic relations “big O” and “small o”, we consider the Cauchy sums of the form ∑a≤k 0 ; in the case where α= 1 these are strictly related to the celebrated Riemann sums. After having learned how to approximate such sums, we apply the results to the approximation of sums of the form ∑0≤
Let’s learn to count
No full textIn this chapter, after a quick review of the basic concepts of set theory, we introduce the fundamental notions and principles of combinatorics. Even though its contents are elementary, we warmly suggest to take a look at the chapter. Our approach consists in trying to describe every combinatorial problem by means of sets of (ordered) sequences, or (unordered) collections, and their dual concepts of sharings and compositions. Computations are successively done via some basic fundamental tools like the Multiplication and the Division Principle. A rigorous and effective formulation of these principles, in particular of the Multiplication Principle, is of fundamental importance for their correct application. Indeed they constitute, at the same time, the royal way to solve combinatorial problems, and the main source of errors, when misused. We conclude the section with a brief discussion of uniform probability on finite sample spaces, which is here just a way to express combinatorial results in probabilistic terms
Counting sequences and collections
No full textIn this chapter we count sequences and sharings, collections and compositions, furnishing many applications and examples. Factorials and binomial coefficients are, on the one hand, indispensable tools for such counting problems, and, on the other hand, their combinatorial interpretation gives a valuable contribution in suggesting and proving many useful identities both concerning sums or alternating sums of binomials and their products
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