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Kovič, and implicitly Ufnarovski and Åhlander, defined a notion of arithmetic partial derivative. We generalize the definition for rational numbers and study several arithmetic partial differential equations of the first and second order. For some equations, we give a complete solution, and for others, we extend previously known results. For example, we… (More)

- Johnathan M. Bardsley, Jorma K. Merikoski, Roberto Vio, Don L. Sturzo
- 2007

The incorporation of nonnegativity constraints in image reconstruction problems is known to have a stabilizing effect on solution methods. In this paper, we both demonstrate and provide an explanation of this phenomena when the image reconstruction problem of interest has least squares form. The benefits of using this natural constraint suggest the… (More)

We present a method, based on series expansions and symmetric polynomials, by which a mean of two variables can be extended to several variables. We apply it mainly to the logarithmic mean.

- Pentti Haukkanen, Jorma K. Merikoski
- Ars Comb.
- 2012

- Pentti Haukkanen, Jorma K. Merikoski
- ArXiv
- 2011

We present an asymptotic formula for the number of line segments connecting q + 1 points of an n × n square grid, and a sharper formula, assuming the Riemann hypothesis. We also present asymptotic formulas for the number of lines through at least q points and, respectively, through exactly q points of the grid. The well-known case q = 2 is so generalized.

- Pentti Haukkanen, Jorma K. Merikoski
- Discrete Applied Mathematics
- 2013

Let m,n ≥ 2, m ≤ n. It is well-known that the number of (twodimensional) threshold functions on an m× n rectangular grid is t(m,n) = 6 π (mn) +O(mn logn) +O(mn log logn) = 6 π (mn) +O(mn logm). We improve the error term by showing that t(m,n) = 6 π (mn) +O(mn).

Given n ∈ Z, its arithmetic derivative n is defined as follows: (i) 0 = 1 = (−1)′ = 0. (ii) If n = up1 · · · pk, where u = ±1 and p1, . . . , pk are primes (some of them possibly equal), then

where suB denotes the sum of the entries of a matrix B and m ≥ 0 (define 0 = 1). Hoffman’s proof was based on certain properties of stochastic matrices. Much later, in 1985, Sidorenko [9], without knowing Hoffman’s work, gave an independent proof as an elementary application of Hölder’s inequality. In 1990, Virtanen [10] generalized (1.1) to the… (More)

The rank subtractivity partial ordering is defined on Cn×n (n ≥ 2) by A ≤− B⇔ rank(B−A) = rankB− rankA, and the star partial ordering by A ≤∗ B⇔ A∗A = A∗B ∧ AA∗ = BA∗. If A and B are normal, we characterize A ≤− B. We also show that then A ≤− B ∧ AB = BA⇔ A ≤∗ B⇔ A ≤− B ∧ A ≤− B. Finally, we remark that some of our results follow from well-known results on… (More)

- JORMA K. MERIKOSKI, ASREY RAJPUT
- 2013

Abstract. Consider a finite, simple, undirected, and bipartite graph G with vertex sets V = {v1, . . . , vm} and W = {w1, . . . , wn}, m ≤ n, V ∩ W = ∅. Let the vertices of V have degrees d1 ≥ d2 ≥ · · · ≥ dm > 0, respectively. Let Ni be the set of neighbors of vi (i = 1, . . . ,m). Define dij = |Ni ∩Nj | (i, j = 1, . . . ,m), where | . | stands for the… (More)