相关论文: Counting real rational functions with all real cri…
We introduce and discuss a new class of (multivalued analytic) transcendental functions which still share with algebraic functions the property that the number of their isolated zeros can be explicitly counted. On the other hand, this class…
Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with…
We study structured optimization problems with polynomial objective function and polynomial equality constraints. The structure comes from a multi-grading on the polynomial ring in several variables. For fixed multi-degrees we determine the…
We express multiplicities and degree functions of graded families of $\mathfrak{m}_R$-primary ideals in an excellent normal local ring $(R,\mathfrak{m}_R)$ as limits of intersection products. Moreover, in dimension 2, we show more refined…
We study the number of meromorphic functions on a Riemann surface with given critical values and prescribed multiplicities of critical points and values. When the Riemann surface is $\CP^1$ and the function is a polynomial, we give an…
In this paper we present a recurrent relation for counting meaningful compositions of the higher-order differential operations on the space $R^{n}$ (n=3,4,...) and extract the non-trivial compositions of order higher than two.
According to the available publications, the field theoretical renormalization group (RG) approach in the two-dimensional case gives the critical exponents that differ from the known exact values. This fact was attempted to explain by the…
In our recent publication we obtained a series expansion of the arctangent function involving complex numbers. In this work we show that this formula can also be expressed as a real rational function.
The study of a machine learning problem is in many ways is difficult to separate from the study of the loss function being used. One avenue of inquiry has been to look at these loss functions in terms of their properties as scoring rules…
Planar commutative n-complex numbers of the form u=x_0+h_1x_1+h_2x_2+...+h_{n-1}x_{n-1} are introduced in an even number n of dimensions, the variables x_0,...,x_{n-1} being real numbers. The planar n-complex numbers can be described by the…
We attempt to quantify the exact proportion of monic $p$-adic polynomials of degree $n$ which are irreducible. We find an exact answer to this when $n$ is prime and $p \neq n$, and also when $n = 4$ and $p \neq 2$. Our answers are rational…
We will show the rationality of the function field of two variables under certain Cremona transformation.
Diagonals of rational functions are an important class of functions arising in number theory, algebraic geometry, combinatorics, and physics. In this paper we study the diagonal grade of a function $f$, which is defined to be the smallest…
A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. This work focuses particularly on periods depending on a parameter: in this case the period…
We consider critical points of a class of functionals on compact four-dimensional manifolds arising from Regularized Determinants for conformally covariant operators, whose explicit form was derived in [10], extending Polyakov's formula.…
We study equations with infinitely many derivatives. Equations of this type form a new class of equations in mathematical physics. These equations originally appeared in p-adic and later in fermionic string theories and their investigation…
Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical…
We give a~detailed construction of the complete ordered field of real numbers by means of infinite decimal expansions. We prove that in the canonical encoding of decimals neither addition nor multiplication is {\em computable}, but that…
We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations, Godel numbering, the fundamental theorem of arithmetic, continued fractions, Egyptian…
We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle: (i) Through the realization matrix of Schur stable systems. (ii) The Blaschke-Potapov product, which is then…