Monotone additive matrix transformations

We investigate additive transformations on the space of real or complex matrices that are monotone with respect to any admissible partial order relation. A complete characterization of these transformations is obtained. In the real case, we show that such transformations are linear and that all nonzero monotone transformations are bijective. As a corollary, we characterize all additive transformations that are monotone with respect to certain classical matrix order relations, in particular, with respect to the Drazin order, left and right *-orders, and the diamond order.     

New differential properties of sup-type functions

In this paper, we study the directional derivative, subderivative, and subdifferential of sup-type functions without any compactness assumption on the index set. As applications, we provide an estimate of the Lipschitz modulus for sup-type functions.     

On differential properties of real functions

We study the categorical analog of the Stepanov theorem and new criteria of asymptotic differentiability of real functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 842–848, July, 1994.     

Monotone insertion of lattice-valued functions

Insertion of lattice-valued functions in a monotone manner is investigated. For L a ⊲-separable completely distributive lattice (i.e. L admits a countable base which is free of supercompact elements), a monotone version of the Katětov-Tong insertion theorem for L-valued functions is established. We also provide a monotone lattice-valued version of Urysohn’s lemma. Both results yield new characterizations of monotonically normal spaces. Moreover, extension of lattice-valued functions under additional assumptions is shown to characterize also monotone normality. This research was supported by the MEyC and FEDER under grant MTM2006-14925-C02-02/ and by UPV05/101     

On the differential properties of continuous functions

We introduce and investigate some new differential properties of continuous functions by means of the geometrical properties of their derivatives.

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Про диференціальні властивості неперервних функцій

Введено та досліджено деякі нові диференціальні властивості неперервних функцій за допомогою геометричних властивостей їх похідних.

     

Monotone and cash-invariant convex functions and hulls

This paper provides some useful results for convex risk measures. In fact, we consider convex functions on a locally convex vector space E which are monotone with respect to the preference relation implied by some convex cone and invariant with respect to some numeraire (‘cash’). As a main result, for any function f, we find the greatest closed convex monotone and cash-invariant function majorized by f. We then apply our results to some well-known risk measures and problems arising in connection with insurance regulation.     

Monotone matrix functions of successive orders

This paper extends a result obtained by Wigner and von Neumann. We prove that a non-constant real-valued function, , in where is an interval of the real line, is a monotone matrix function of order on if and only if a related, modified function is a monotone matrix function of order for every value of in , assuming that is strictly positive on .

     

On differential properties of functions of bounded variation

It is established that Karagulyan??s exact estimate of the divergence rate of strong integral means of summable functions is extendable to strong means of additive functions of intervals having bounded variation. Furthermore, it is proved that each function defined on [0, 1] ⁿ with bounded variation in the sense of Hardy has a strong gradient at almost every point (this strengthens the corresponding result of Burkill and Haslam-Jones on the differentiability almost everywhere), whereas the same is not true for functions with bounded variation in the sense of Arzela.     

Monotone thematic factorizations of matrix functions

We continue the study of the so-called thematic factorizations of admissible very badly approximable matrix functions. These factorizations were introduced by V.V. Peller and N.J. Young for studying superoptimal approximation by bounded analytic matrix functions. Even though thematic indices associated with a thematic factorization of an admissible very badly approximable matrix function are not uniquely determined by the function itself, R.B. Alexeev and V.V. Peller showed that the thematic indices of any monotone non-increasing thematic factorization of an admissible very badly approximable matrix function are uniquely determined. In this paper, we prove the existence of monotone non-decreasing thematic factorizations for admissible very badly approximable matrix functions. It is also shown that the thematic indices appearing in a monotone non-decreasing thematic factorization are not uniquely determined by the matrix function itself. Furthermore, we show that the monotone non-increasing thematic factorization gives rise to a great number of other thematic factorizations.     

Monotone trajectories of differential inclusions and functional differential inclusions with memory

The paper gives a necessary and sufficient condition for the existence of monotone trajectories to differential inclusionsdx/dt ∈S[x(t)] defined on a locally compact subsetX ofR ^p, the monotonicity being related to a given preorder onX. This result is then extended to functional differential inclusions with memory which are the multivalued case to retarded functional differential equations. We give a similar necessary and sufficient condition for the existence of trajectories which reach a given closed set at timet=0 and stay in it with the monotonicity property fort≧0.     

Monotone Boolean functions capture their primes

It is shown that monotone Boolean functions on the Boolean cube capture the expected number of primes, under the usual identification by binary expansion. This answers a question posed by G. Kalai.     

Monotone additive functions on shifted primes

Generalising a sixty year old result of Erdös, it is proved that an additive arithmetic function that is non-decreasing on the shifted primes is essentially a logarithm.     

Monotone solutions of second-order nonlinear differential equations

Necessary and sufficient conditions are obtained for existence of monotone solutions of a nonlinear differential equation. As applications, several existence criteria and comparison theorems are derived.     

Monotone iterative technique for delay differential equations

In proving existence of extremal solutions for delay differential equations, one usually assumes nondecreasing property on the function involved without which the proof breaks down. Our main purpose, in this paper, is to remove the monotone assumption by proving a new comparison result which is required in the analysis and which avoids the standard line of argument. Furthermore, our comparison theorem shows that the choice of initial functions cannot be arbitrary for the results to hold. Using this result we then develop monotone technique to obtain extremal solutions.