Riemann Zeta Based Surge Modelling of Continuous Real Functions in Electrical Circuits

Riemann zeta is defined as a function of a complex variable that analytically continues the sum of the Dirichlet series, when the real part is greater than unity. In this paper, the Riemann zeta associated with the finite energy possessed by a 2mm radius, free falling water droplet, crashing into a plane is considered. A modified zeta function is proposed which is incorporated to the spherical coordinates and real analysis has been performed. Through real analytic continuation, the single point of contact of the drop at the instant of touching the plane is analyzed. The zeta function is extracted at the point of destruction of the drop, where it defines a unique real function. A special property is assumed for some continuous functions, where the function’s first derivative and first integral combine together to a nullity at all points. Approximate reverse synthesis of such a function resulted in a special waveform named the dying-surge. Extending the proposed concept to general continuous real functions resulted in the synthesis of the corresponding function’s Dying-surge model. The Riemann zeta function associated with the water droplet can also be modeled as a dying–surge. The Dying- surge model corresponds to an electrical squeezing or compression of a waveform, which was originally defined over infinite arguments, squeezed to a finite number of values for arguments placed very close together with defined final and penultimate values. Synthesized results using simulation software are also presented, along with the analysis. The presence of surges in electrical circuits will correspond to electrical compression of some unknown continuous, real current or voltage function and the method can be used to estimate the original unknown function.

Deminutywność, symilatywność i tautologia słowotwórcza w języku doby średniopolskiej (na materiale ze słownika M.A. Troca)

W tekście podejmowana jest problematyka ograniczeń procesu kategoryzacji klas derywatów deminutywnych oraz symilatywnych w dobie średniopolskiej. Celem opracowania było wskazanie potencjalnych przyczyn blokowania procesów kategoryzacyjnych klas historycznych deminutywów oraz symilatywów. W zakresie metodologii i ustaleń terminologicznych wykorzystano osiągnięcia tzw. „katowickiej szkoły słowotwórstwa historycznego”. Głównym źródłem materiału leksykalnego stał się trójjęzyczny dykcjonarz M.A. Troca z 1764 roku (jego III tom, z polszczyzną jako językiem wyjściowym). Świadomość lingwistyczna autora tego słownika, przejawiająca się w sposobie organizacji wyrażeń hasłowych oraz doboru ekwiwalentów wraz z definicjami, rzuciła nowe światło na sposób identyfikowania kategorii deminutywów, symilatywów, a także formacji tautologicznych przez dawnych użytkowników języka. Okazało się, że w drugiej połowie XVIII wieku żadna z tych klas nie wykrystalizowała swoich dominant, zaś czynnikiem, który mógł podtrzymywać ten stan, była obecność w języku znacznej liczby derywatów tautologicznych względem podstawy, budowanych z udziałem wielofunkcyjnych formantów z podstawowymi sufiksalnymi spółgłoskami -k- i -c-. Diminutivity, similativity and word-formation tautology in Middle Polish (illustrated with data from M.A. Troc’s Dictionary) Summary: The text deals with the limitations of the categorization process of the classes of diminutive and similative derivatives in Middle Polish. The aim of the study was to identify the potential reasons for the blocking of the categorization processes of the historical classes of diminutives and similatives. The methodology and terminology used in the paper follows the achievements of the so-called “Katowice school of historical word-formation”. The 1764 trilingual dictionary by M.A. Troc (Volume 3, with Polish as the input language) was the main source of lexical material. Based on the analysis of the presented material, one can conclude that the linguistic awareness of the lexicographer, manifested through the organization of dictionary entries and the choice of foreign equivalents and their definitions, may shed a new light on the categorical system of historical derivatives. In lack of sufficient Polish-language contexts, the translational character of lexicographic sources lets us gain information about the semantic and stylistic value of Polish lexical units on the basis of their foreign equivalents or their foreign-language definitions provided by dictionaries. The category of diminutive names in the second half of the 18th century did not yet crystallize its dominants, and the class of similative names had a similar formal and semantic status. Both classes constituted products of sets that contained derivative units, assuming a diminishing or similative function. The factor that inhibited the process of the crystallization of the dominants in the mentioned classes was the extremely high level of word-formation tautology, which did not allow language users to identify the real functions of multifunctional formants with the basic consonants -k- and -c-.

On the Representation of Boolean and Real Functions as Hamiltonians for Quantum Computing

Mapping functions on bits to Hamiltonians acting on qubits has many applications in quantum computing. In particular, Hamiltonians representing Boolean functions are required for applications of quantum annealing or the quantum approximate optimization algorithm to combinatorial optimization problems. We show how such functions are naturally represented by Hamiltonians given as sums of Pauli Z operators (Ising spin operators) with the terms of the sum corresponding to the function’s Fourier expansion. For many classes of Boolean functions which are given by a compact description, such as a Boolean formula in conjunctive normal form that gives an instance of the satisfiability problem, it is #P-hard to compute its Hamiltonian representation, i.e., as hard as computing its number of satisfying assignments. On the other hand, no such difficulty exists generally for constructing Hamiltonians representing a real function such as a sum of local Boolean clauses each acting on a fixed number of bits as is common in constraint satisfaction problems. We show composition rules for explicitly constructing Hamiltonians representing a wide variety of Boolean and real functions by combining Hamiltonians representing simpler clauses as building blocks, which are particularly suitable for direct implementation as classical software. We further apply our results to the construction of controlled-unitary operators, and to the special case of operators that compute function values in an ancilla qubit register. Finally, we outline several additional applications and extensions of our results to quantum algorithms for optimization. A goal of this work is to provide a design toolkit for quantum optimization which may be utilized by experts and practitioners alike in the construction and analysis of new quantum algorithms, and at the same time to provide a unified framework for the various constructions appearing in the literature.

The Space of Functions with Tempered Increments on a Locally Compact and Countable at Infinity Metric Space

The aim of the paper is to introduce the Banach space consisting of real functions defined on a locally compact and countable at infinity metric space and having increments tempered by a modulus of continuity. We are going to provide a condition that is sufficient for the relative compactness in the Banach space in question. A few particular cases of that Banach space will be discussed.

The Weyl–Mellin quantization map

In this paper, we present a quantization of the functions of spacetime, i.e. a map, analog to Weyl map, which reproduces the [Formula: see text]-Minkowski commutation relations, and it has the desirable properties of mapping square integrable functions into Hilbert–Schmidt operators, as well as real functions into symmetric operators. The map is based on Mellin transform on radial and time coordinates. The map also defines a deformed ∗ product which we discuss with examples.

Solution of Linear Fuzzy Fractional Differential Equations Using Fuzzy Natural Transform

The purpose of this paper is to apply the fuzzy natural transform (FNT) for solving linear fuzzy fractional ordinary differential equations (FFODEs) involving fuzzy Caputo’s H-difference with Mittag-Leffler laws. It is followed by proposing new results on the property of FNT for fuzzy Caputo’s H-difference. An algorithm was then applied to find the solutions of linear FFODEs as fuzzy real functions. More specifically, we first obtained four forms of solutions when the FFODEs is of order α∈(0,1], then eight systems of solutions when the FFODEs is of order α∈(1,2] and finally, all of these solutions are plotted using MATLAB. In fact, the proposed approach is an effective and practical to solve a wide range of fractional models.

Real Functions, Covers and Bornologies

The paper tries to survey the recent results about relationships between covering properties of a topological space X and the space USC(X) of upper semicontinuous functions on X with the topology of pointwise convergence. Dealing with properties of continuous functions C(X), we need shrinkable covers. The results are extended for A-measurable and upper A-semimeasurable functions where A is a family of subsets of X. Similar results for covers respecting a bornology and spaces USC(X) or C(X) endowed by a topology defined by using the bornology are presented. Some of them seem to be new.