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Multivariable, real-valued functions induce matrix-valued functions defined on the space of d-tuples of n-times-n pairwise-commuting self-adjoint matrices. We examine the geometry of this space of matrices and conclude that the best notion…
Validation is a major challenge in differentiable programming. The state of the art is based on algorithmic differentiation. Consistency of first-order tangent and adjoint programs is defined by a well-known first-order differential…
This paper is concerned with the directional derivative of the value function for a very general set-constrained optimization problem under perturbation. Under reasonable assumptions, we obtain upper and lower estimates for the upper and…
For non-anticipative functionals, differentiable in Chitashvili's sense, the It\^o formula for cadlag semimartingales is proved. Relations between different notions of functional derivatives are established.
We show that irreducibility is not a first-order definable property of real algebraic varieties. The proof is based on the recent o-minimality result for the exponential function. We conjecture that irreducibility is not a definable…
We explain the exact meaning of a statement we made in a previous paper on invariants, namely that a complex-valued function of the data of the functional equation of an $L$-function is an invariant if and only if it is stable under the…
Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex…
We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation".…
This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar…
As established by R T. Rockafellar, real valued convex-concave functions are generically differentiable. It this paper we shall show that for a convex-concave function defined on an open convex set $C \times D,$ there exist dense subsets…
The change of variable theorem is proved under the sole hypothesis of differentiability of the transformation. Specifically, it is shown under this hypothesis that the transformed integral equals the given one over every measurable subset…
Theory of differential subordination provides techniques to reduce differential subordination problems into verifying some simple algebraic condition called admissibility condition. We exploit the first order differential subordination…
We discuss some surprising phenomena from basic calculus related to oscillating functions and to the theorem on the differentiability of inverse functions. Among other things, we see that a continuously differentiable function with a strict…
This paper proves that in Size Theory the comparison of multidimensional size functions can be reduced to the 1-dimensional case by a suitable change of variables. Indeed, we show that a foliation in half-planes can be given, such that the…
The notion of a real-valued function is central to mathematics, computer science, and many other scientific fields. Despite this importance, there are hardly any positive results on decision procedures for predicate logical theories that…
We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained…
The results presented in this paper are refinements of some results presented in a previous paper. Three such refined results are presented. The first one relaxes one of the basic hypotheses assumed in the previous paper, and thus extends…
In this paper we determine the region of variability for certain subclasses of univalent functions satisfying differential inequalities. In the final section we graphically illustrate the region of variability for several sets of…
Some sufficient conditions on certain constants which are involved in some first, second and third order differential subordinations associated with certain functions with positive real part like modified Sigmoid function, exponential…
We present some properties of the gradient of a mu-differentiable function. The Method of Lagrange Multipliers for mu-differentiable functions is then exemplified.