Easy Proof Research Articles

On the bordification of Outer space

We give a simple construction of an equivariant deformation retract of Outer space which is homeomorphic to the Bestvina-Feighn bordification. This results in a much easier proof that the bordification is (2n-5)-connected at infinity, and hence that $Out(F_n)$ is a virtual duality group.

Axioms for the fixed point index of n-valued maps, and some applications

We give an axiomatic characterization of the fixed point index of an n-valued map. For n-valued maps on a polyhedron, the fixed point index is shown to be unique with respect to axioms of homotopy invariance, additivity, and a splitting property. This uniqueness is used to obtain easy proofs of an averaging formula and product formula for the index. In the setting of n-valued maps on a manifold, we show that the axioms can be weakened.

Projectively Invariant Objects and the Index of the Group of Affine Transformations in the Group of Projective Transformations

The paper is grown from the lecture course "Metric projective geometry" which I hold at the summer school "Finsler geometry with applications" at Karlovassi, Samos, in 2014, and at the workshop before the 8th seminar on Geometry and Topology of the Iranian Mathematical society at the Amirkabir University of Technology in 2015. The goal of this lecture course was to show how effective projectively invariant objects can be used to solve natural and named problems in differential geometry, and this paper also does it: I give easy new proofs to many known statements, and also prove the following new statement: on a complete Riemannian manifold of nonconstant curvature the index of the group of affine transformations in the group of projective transformations is at most two.

Algorithmic and algebraic aspects of unshuffling permutations

A permutation is said to be a square if it can be obtained by shuffling two order-isomorphic patterns. The definition is intended to be the natural counterpart to the ordinary shuffle of words and languages. In this paper, we tackle the problem of recognizing square permutations from both the point of view of algebra and algorithms. On the one hand, we present some algebraic and combinatorial properties of the shuffle product of permutations. We follow an unusual line consisting in defining the shuffle of permutations by means of an unshuffling operator, known as a coproduct. This strategy allows to obtain easy proofs for algebraic and combinatorial properties of our shuffle product. We besides exhibit a bijection between square (213,231)-avoiding permutations and square binary words. On the other hand, by using a pattern avoidance criterion on directed perfect matchings, we prove that recognizing square permutations is NP-complete.

On a (No Longer) New Segal Algebra: A Review of the Feichtinger Algebra

Since its invention in 1979 the Feichtinger algebra has become a useful Banach space of functions with applications in time-frequency analysis, the theory of pseudo-differential operators and several other topics. It is easily defined on locally compact Abelian groups and, in comparison with the Schwartz(-Bruhat) space, the Feichtinger algebra allows for more general results with easier proofs. This review paper develops the theory of Feichtinger’s algebra in a linear and comprehensive way. The material gives an entry point into the subject and it will also bring new insight to the expert. A further goal of this paper is to show the equivalence of the many different characterizations of the Feichtinger algebra known in the literature. This task naturally guides the paper through basic properties of functions that belong to this space, over operators on it, and to aspects of its dual space. Additional results include a seemingly forgotten theorem by Reiter on Banach space isomorphisms of the Feichtinger algebra, a new identification of Feichtinger’s algebra as the unique Banach space in $$L^{1}$$ with certain properties, and the kernel theorem for the Feichtinger algebra. A historical description of the development of the theory, its applications, and a list of related function space constructions is included.

A transfer principle for second-order arithmetic, and applications

In the theory of conditional sets, many classical theorems from areas such as analysis, probability theory or measure theory are lifted to the conditional framework, often to be applied in areas such as mathematical economics or optimization. The frequent experience that such theorems can be proved by `conditionalizations' of the classical proofs suggests that a general transfer principle is in the background, and that formulating and proving such a transfer principle would yield a wealth of useful further conditional versions of classical results, in addition to providing a uniform approach to the results already known. In this paper, we formulate and prove such a transfer principle based on second-order arithmetic, which, by the results of reverse mathematics, suffices for the bulk of classical mathematics, including real analysis, measure theory and countable algebra, and excluding only more remote realms like category theory, set-theoretical topology or uncountable set theory, see e.g. the introduction of \cite{simpson2009subsystems}. This transfer principle is then employed to give short and easy proofs of conditional versions of central results in various areas of mathematics, including the Bolzano-Weierstrass theorem, the Heine-Borel theorem, the Riesz representation theorem and Brouwer's fixed point theorem.

LEADING PRINCIPAL MINORS AND SEMIDEFINITENESS

Semidefinite matrices often arise in economic models, usually as Hessian matrices of convex or concave functions. Anytime the matrix can be semidefinite, rather than definite, the task of characterizing it is burdensome because extant results require that all principal minors be signed. A theorem is presented that shows it is sufficient to sign only selected principal minors when the matrix has a definite submatrix. This theorem is particularly useful in duality applications. The theorem also provides relatively easy proof of the standard relationship between semidefiniteness and principal minors. (JEL C02)

THE N\xc9RON\u2013SEVERI LIE ALGEBRA OF A SOERGEL MODULE

We introduce the Néron–Severi Lie algebra of a Soergel module and we determine it for a large class of Schubert varieties. This is achieved by investigating which Soergel modules admit a tensor decomposition. We also use the Néron–Severi Lie algebra to provide an easy proof of the well-known fact that a Schubert variety is rationally smooth if and only if its Betti numbers satisfy Poincaré duality.

Constructing cospectral graphs via a new form of graph product

We present a new method to construct a family of co-spectral graphs. Our method is based on a new type of graph product that we define, the bipartite graph product, which may be of self-interest. Our method is different from existing techniques in the sense that it is not based on a sequence of local graph operations (e.g. Godsil–McKay switching). The explicit nature of our construction allows us, for example, to construct an infinite family of cospectral graphs and provide an easy proof of non-isomorphism. We are also able to characterize fully the spectrum of the cospectral graphs.

Automorphism groups of finitely generated groups that are isomorphic to group of inner automorphisms

Let G be a finitely generated group and let C* denote the group of all central automorphisms of G fixing the center of G elementwise. Azhdari and Malayeri [J. Algebra Appl. 6 (2011), 1283–1290] gave necessary and sufficient conditions on G such that C* ≃ Inn(G). We prove a more general technical lemma and, as a consequence, obtain a short and easy proof of this result of Azhdari and Malayeri and generalize their result; and also generalize the main result of Nasrabadi and Farimani [Indag. Math. (N. S.) 26 (2015), 137–141] from finite autonilpotent p-groups of class 2 to arbitrary finitely generated nilpotent groups of class 2. Subsequently, we also obtain short proofs of some other existing and some new related results.

Existence and uniqueness of solutions to the second order fuzzy dynamic equations on time scales

In this paper, we introduce a new metric space to study the existence and uniqueness of solutions to second order fuzzy dynamic equations on time scales. In this regard, we use Banach’s fixed point theorem to prove this result. Also, we see that this metric guarantees an elegant and easier proof for the existence of solutions to second order fuzzy dynamic equations on time scales.

A Very Easy Proof of Arrow's Impossibility Theorem Using Social Welfare Functionals

In this paper we provide Impossibility Theorems for social welfare functionals. We adapt the method of proving the inevitability of dictatorial Bergson-Samuelson social welfare functions for fixed utility profiles developed in a paper by Robert P. Parks in 1976. With a little more from results already available in the literature we are able to prove impossibility under conditions on the admissible set which are milder than the ones existing in the literature.

Korovkin theorems on weighted spaces: revisited

In this note, using the idea of the late Professor Gadjiev (Mat Zametki 20(5):781–786, 1976), we give new, direct and easy proofs of the Korovkin theorems for positive linear operators acting on weighted spaces. Recent improvements and new applications are also presented.

Solving Smullyan Puzzles with Formal Systems

Solving numeric, logic and language puzzles and paradoxes is common within a wide community of high school and university students, fact witnessed by the increasing number of books published by mathematicians such as Martin Gardner (popular books as old as Gardner in Aha! insight. W. H. Freeman & Co., London, 1978, Wheels, life and other mathematical amusements. W H Freeman & Co., London, 1985), Douglas Hofstadter [in one of the best popular science books on paradoxes (Hofstadter in Godel, escher, bach: an eternal golden braid, Penguin, London, 2000)], inspired by Godel’s incompleteness theorems), Patrick Hughes and George Brecht (see Hughes and Brecht in Vicious circles and infinity, an anthology of paradoxes. Penguin Books, London, 1993) and Raymond M. Smullyan (the most well known being Smullyan in Forever undecided, puzzle guide to godel. Oxford Paperbacks, Oxford 1988, To Mock a Mockingbird and other logic puzzles. Oxford Paperbacks, Oxford 2000, The lady or the tiger? And other logic puzzles. Dover Publications Inc., Mineola 2009), inter alia. Books by Smullyan (such as Smullyan 1988, 2000) are, however, much more involved, since they introduce learning trajectories and strategies across several subjects of mathematical logic, as difficult as combinatorial logic (see, e.g., Smullyan 2000), computability theory (see Smullyan 1988), and proof theory (see Smullyan 1988, 2009). These books provide solutions to their suggested exercises. Both statements and their solutions are written in the natural language, introducing some informal algorithms. As an exercise in Mathematics we wonder if an easy proof system could be devised to solve the amusing equations proposed by Smullyan in his books. Moreover, university students of logic could well train themselves in constructing deductive systems to solve puzzles instead of a non-uniform treatment one by one. In this paper, addressing students, we introduce one such formal systems, a tableaux approach able to provide the solutions to the puzzles involving either propositional logic, first order logic, or aspect logic. Let the reader amuse herself or himself!

Normes cyclotomiques naïves et unités logarithmiques

We compute the \({\mathbb {Z}}\)-rank of the subgroup \(\widetilde{E}_K =\bigcap _{n\in {\mathbb {N}}} N_{K_n/K}(K_n^\times )\) of elements of the multiplicative group of a number field K that are norms from every finite level of the cyclotomic \({\mathbb {Z}}_\ell \)-extension \(K^c\) of K. Thus we compare its \(\ell \)-adification \({\mathbb {Z}}_\ell \otimes _{\mathbb {Z}}\widetilde{E}_K\) with the group of logarithmic units \(\widetilde{\varepsilon }_K\). By the way we point out an easy proof of the Gross–Kuz’min conjecture for \(\ell \)-undecomposed extensions of abelian fields.

Implementability by a Canonical Indirect Mechanism of an Optimal Two-Dimensional Direct Mechanism

The present paper investigates the multi-dimensional mechanism design in which buyers have taste and budget as their private information. The paper shows an easy proof of a two-dimensional optimal direct mechanism by a one-dimensional indirect mechanism: A canonical mechanism in the traditional one-dimensional setting, i.e., function of one variable, the buyer’s taste. It also sheds light on where the difficulty lies implementability of a general direct mechanism—not optimal—by a canonical mechanism.

Genericity Results in Linear Conic Programming—A Tour d’Horizon

This paper is concerned with so-called generic properties of general linear conic programs. Many results have been obtained on this subject during the last two decades. For example, it is known that uniqueness, strict complementarity, and nondegeneracy of optimal solutions hold for almost all problem instances. Strong duality holds generically in a stronger sense, i.e., it holds for a generic subset of problem instances. In this paper, we survey known results and present new ones. In particular we give an easy proof of the fact that Slater’s condition holds generically in linear conic programming. We further discuss the problem of stability of uniqueness, nondegeneracy, and strict complementarity. We also comment on the fact that in general, a conic program cannot be treated as a smooth problem and that techniques from nonsmooth geometric measure theory are needed.

Loop near-rings and unique decompositions of H-spaces

For every H-space $X$ the set of homotopy classes $[X,X]$ possesses a natural algebraic structure of a loop near-ring. Albeit one cannot say much about general loop near-rings, it turns out that those that arise from H-spaces are sufficiently close to rings to have a viable Krull--Schmidt type decomposition theory, which is then reflected into decomposition results of H-spaces. In the paper we develop the algebraic theory of local loop near-rings and derive an algebraic characterization of indecomposable and strongly indecomposable H-spaces. As a consequence, we obtain unique decomposition theorems for products of H-spaces. In particular, we are able to treat certain infinite products of H-spaces, thanks to a recent breakthrough in the Krull--Schmidt theory for infinite products. Finally, we show that indecomposable finite $p$-local H-spaces are automatically strongly indecomposable, which leads to an easy alternative proof of classical unique decomposition theorems of Wilkerson and Gray.

Mixed-norm estimates and symmetric geometric means

The mixed-norm versions of the H\"older and Minkowski integral inequalities are used to produce new, general estimates involving symmetric geometric means of mixed norms. Various existing mixed-norm estimates are shown to be simple special cases of these new results. Examples are also given of applying mixed-norm H\"older and Minkowski to other estimates, providing much easier proofs. Finally, the effectiveness of this technique is demonstrated by deriving a new inequality which combines features from two separate previous results.

The Lidskii trace property and the nest approximation property in Banach spaces

For a Banach space X, the Lidskii trace property is equivalent to the nest approximation property; that is, for every nuclear operator on X that has summable eigenvalues, the trace of the operator is equal to the sum of the eigenvalues if and only if for every nest N of closed subspaces of X, there is a net of finite rank operators on X, each of which leaves invariant all subspaces in N, that converges uniformly to the identity on compact subsets of X. The Volterra nest in Lp(0,1), 1≤p<∞, is shown to have the Lidskii trace property. A simpler duality argument gives an easy proof of the result [2, Theorem 3.1] that an atomic Boolean subspace lattice that has only two atoms must have the strong rank one density property.