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On some Zarankiewicz numbers and bipartite Ramsey Numbers for Quadrilateral
PublicationThe Zarankiewicz number z ( m, n ; s, t ) is the maximum number of edges in a subgraph of K m,n that does not contain K s,t as a subgraph. The bipartite Ramsey number b ( n 1 , · · · , n k ) is the least positive integer b such that any coloring of the edges of K b,b with k colors will result in a monochromatic copy of K n i ,n i in the i -th color, for some i , 1 ≤ i ≤ k . If n i = m for all i , then we denote this number by b k ( m )....
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Polyhedral Ramsey Numbers
PublicationGiven two polygons or polyhedrons P1 and P2, we can transform these figures to graphs G1 and G2, respectively. The polyhedral Ramsey number Rp(G1,G2) is the smallest integer n such that every graph, which represents polyhedron on n vertices either contains a copy of G1 or its complement contains a copy of G2. Using a computer search together with some theoretical results we have established some polyhedral Ramsey numbers, for example...
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Shannon Capacity and Ramsey Numbers
PublicationRamsey-type theorems are strongly related to some results from information theory. In this paper we present these relations.
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Turán numbers for odd wheels
PublicationThe Turán number ex(n,G) is the maximum number of edges in any n-vertex graph that does not contain a subgraph isomorphic to G. A wheel W_n is a graph on n vertices obtained from a C_{n−1} by adding one vertex w and making w adjacent to all vertices of the C_{n−1}. We obtain two exact values for small wheels: ex(n,W_5)=\lfloor n^2/4+n/2\rfloor, ex(n,W_7)=\lfloor n^2/4+n/2+1 \rfloor. Given that ex(n,W_6) is already known, this...
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Generating sequences of Lefschetz numbers of iterates
PublicationDu, Huang and Li showed in 2003 that the class of Dold–Fermat sequences coincides with the class of Newton sequences, which are defined in terms of socalled generating sequences. The sequences of Lefschetz numbers of iterates form an important subclass of Dold–Fermat (thus also Newton) sequences. In this paper we characterize generating sequences of Lefschetz numbers of iterates.
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On the connected and weakly convex domination numbers
PublicationIn this paper we study relations between connected and weakly convex domination numbers. We show that in general the difference between these numbers can be arbitrarily large and we focus on the graphs for which a weakly convex domination number equals a connected domination number. We also study the influence of the edge removing on the weakly convex domination number, in particular we show that a weakly convex domination number...
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A NOTE ON ON-LINE RAMSEY NUMBERS FOR QUADRILATERALS
PublicationWe consider on-line Ramsey numbers defined by a game played between two players, Builder and Painter. In each round Builder draws an the edge and Painter colors it either red or blue, as it appears. Builder’s goal is to force Painter to create a monochromatic copy of a fixed graph H in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number \widetilde{r}(H) of...
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On-line Ramsey Numbers of Paths and Cycles
PublicationConsider a game played on the edge set of the infinite clique by two players, Builder and Painter. In each round, Builder chooses an edge and Painter colours it red or blue. Builder wins by creating either a red copy of $G$ or a blue copy of $H$ for some fixed graphs $G$ and $H$. The minimum number of rounds within which Builder can win, assuming both players play perfectly, is the \emph{on-line Ramsey number} $\tilde{r}(G,H)$. In...
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Distance paired domination numbers of graphs
PublicationW pracy przedstawione są pewne własności liczb k-dominowania parami w grafach. Wykazane jest, że problem decyzyjny liczby k-dominowania parami jest problemem NP-zupełnym nawet dla grafów dwudzielnych. Przedstawione są ograniczenia górne i dolne dla liczby k-dominowania parami w drzewach i scharakteryzowane drzewa, w których te ograniczenia są osiągnięte.
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Total restrained domination numbers of trees
PublicationOpisane są wszystkie drzewa, w których liczby dominowania totalnego i totalno - powściągniętego są sobie równe, a także podano dolne ograniczenie na liczbę dominowania totalno - powściągniętego w drzewach.
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Weakly connected domination subdivision numbers
PublicationLiczba podziału krawędzi dla dominowania słabo spójnego to najmniejsza liczba krawędzi jaką należy podzielić, aby wzrosła liczba dominowania słabo wypukłego. W pracy przedstawione są własności liczby podziału krawędzi dla dominowania słabo spójnego dla różnych grafów.
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Weakly convex and convex domination numbers.
PublicationW artykule przedstawione są nowo zdefiniowane liczby dominowania wypukłego i słabo wypukłego oraz ich porównanie z innymi liczbami dominowania. W szczególności, rozważana jest równość liczby dominowania spójnego i wypukłego dla grafów kubicznych.
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On some open questions for Ramsey and Folkman numbers
PublicationWe discuss some of our favorite open questions about Ramsey numbers and a related problem on edge Folkman numbers. For the classical two-color Ramsey numbers, we first focus on constructive bounds for the difference between consecutive Ramsey numbers. We present the history of progress on the Ramsey number R(5,5) and discuss the conjecture that it is equal to 43.
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Graphs with equal domination and certified domination numbers
PublicationA setDof vertices of a graphG= (VG,EG) is a dominating set ofGif every vertexinVG−Dis adjacent to at least one vertex inD. The domination number (upper dominationnumber, respectively) ofG, denoted byγ(G) (Γ(G), respectively), is the cardinality ofa smallest (largest minimal, respectively) dominating set ofG. A subsetD⊆VGis calleda certified dominating set ofGifDis a dominating set ofGand every vertex inDhas eitherzero...
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Paired domination subdivision and multisubdivision numbers of graphs
PublicationThe paired domination subdivision number sdpr(G) of a graph G is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the paired domination number of G. We prove that the decision problem of the paired domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the paired domination muttisubdivision number of a nonempty graph...
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Total outer-connected domination numbers of trees
PublicationNiech G=(V,E) będzie grafem bez wierzchołków izolowanych. Zbiór wierzchołków D nazywamy zbiorem dominującym totalnym zewnętrznie spójnym jeżli każdy wierzchołek grafu ma sąsiada w D oraz podgraf indukowany przez V-D jest grafem spójnym. Moc najmniejszego zbioru D o takich własnościach nazywamy liczbą dominowania totalnego zewnątrznie spójnego. Praca m.in. zawiera dolne ograniczenie na liczbę dominowania totalnego zewnętrznie spójnego...
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Domination subdivision and domination multisubdivision numbers of graphs
PublicationThe domination subdivision number sd(G) of a graph G is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number of G. It has been shown [10] that sd(T)<=3 for any tree T. We prove that the decision problem of the domination subdivision number is NP-complete even for bipartite graphs. For this reason we define the domination multisubdivision number...
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On some ramsey and turan-type numbers for paths and cycles
PublicationUdowodniono, że R(P_3,C_k,C_k)= R(C_k,C_k)= 2k - 1, dla nieparzystych k. Udowodniono, że R(P_4,P_4,C_k) = k + 2 oraz R(P_3,P_5,C_k) = k + 1 dla k > 2.
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Gene copy numbers of HER family in breast cancer
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The paired-domination and the upper paired-domination numbers of graphs
PublicationIn this paper we obtain the upper bound for the upper paired-domination number and we determine the extremal graphs achieving this bound. Moreover we determine the upper paired- domination number for cycles.
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Graphs with equal domination and 2-distance domination numbers
PublicationW publikacji scharakteryzowane są wszystkie te drzewa i grafy jednocykliczne, w których liczba dominowania oraz liczba 2-dominowania na odległość są sobie równe.
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A note on the weakly convex and convex domination numbers of a torus
PublicationW pracy określone są liczby liczby dominowania i dominowania wypukłego torusów, czyli iloczynów kartezjańskich dwóch cykli.
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Ramsey numbers for triangles versus almost-complete graphs.
PublicationPokazano, że w każdym krawędziowym pokolorowaniu dwoma kolorami grafu pełnego o 38 wierzchołkach występuje trójkąt w pierwszym kolorze lub podgraf izomorficzny z K_10 - e w drugim kolorze. Stąd otrzymujemy górne oszacowanie R(K_3, K_10 - e) <= 38. Przedstawiamy także pokolorowanie krawędziowe grafu K_36, którego istnienie dowodzi, że R(K_3, K_10 - e) >= 37.
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Domination numbers in graphs with removed edge or set of edges
PublicationW artykule przedstawiony jest wpływ usuwania krawędzi lub zbioru krawędzi na liczby dominowania spójnego i słabo spójnego.
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P-Adic Numbers Ultrametric Analysis and Applications
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Scaling of numbers in residue arithmetic with the flexible selection of scaling factor
PublicationA scaling technique of numbers in resudue arithmetic with the flexible selection of the scaling factor is presented. The required scaling factor can be selected from the set of moduli products of the Residue Number System (RNS) base. By permutation of moduli of the number system base it is possible to create many auxilliary Mixed-Radix Systems associated with the given RNS with respect to the base, but they have different sets...
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On trees with equal domination and total outer-independent domination numbers
PublicationFor a graph G=(V,E), a subset D subseteq V(G) is a dominating set if every vertex of V(G)D has a neighbor in D, while it is a total outer-independent dominating set if every vertex of G has a neighbor in D, and the set V(G)D is independent. The domination (total outer-independent domination, respectively) number of G is the minimum cardinality of a dominating (total outer-independent dominating, respectively) set of G. We characterize...
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Weakly convex and convex domination numbers of some products of graphs
PublicationIf $G=(V,E)$ is a simple connected graph and $a,b\in V$, then a shortest $(a-b)$ path is called a $(u-v)$-{\it geodesic}. A set $X\subseteq V$ is called {\it weakly convex} in $G$ if for every two vertices $a,b\in X$ exists $(a-b)$- geodesic whose all vertices belong to $X$. A set $X$ is {\it convex} in $G$ if for every $a,b\in X$ all vertices from every $(a-b)$-geodesic belong to $X$. The {\it weakly convex domination number}...
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Trees with equal restrained domination and total restrained domination numbers
PublicationW publikacji scharakteryzowano wszystkie drzewa, w których liczby dominowania powściągniętego oraz podwójnie totalnego są sobie równe.
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Pipelined division of signed numbers with the use of residue arithmetic in FPGA
PublicationAn architecture of a pipelined signed residue divider for small number ranges is presented. The divider makes use of the multiplicative division algorithm where initially the reciprocal of the divisor is calculated and subsequently multiplied by the dividend. The divisor represented in the signed binary form is used to compute the approximated reciprocal in the residue form by the table look-up. In order to reduce the needed length...
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On trees with equal 2-domination and 2-outer-independent domination numbers
PublicationFor a graph G = (V,E), a subset D \subseteq V(G) is a 2-dominating set if every vertex of V(G)\D$ has at least two neighbors in D, while it is a 2-outer-independent dominating set if additionally the set V(G)\D is independent. The 2-domination (2-outer-independent domination, respectively) number of G, is the minimum cardinality of a 2-dominating (2-outer-independent dominating, respectively) set of G. We characterize all trees...
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FPGA computation of magnitude of complex numbers using modified CORDIC algorithm
PublicationIn this work we present computation of the magnitude of complex numbers using a modified version of the CORDIC algorithm that uses only five iterations. The relationship between the computation error and the number of CORDIC iterations are presented for floating-point and integer arithmetics. The proposed modification of CORDIC for integer arithmetic relies upon the introduction of correction once basic computations are performed...
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Unicyclic graphs with equal total and total outer-connected domination numbers
PublicationLet G = (V,E) be a graph without an isolated vertex. A set D ⊆ V (G) is a total dominating set if D is dominating and the in- duced subgraph G[D] does not contain an isolated vertex. The total domination number of G is the minimum cardinality of a total domi- nating set of G. A set D ⊆ V (G) is a total outer–connected dominating set if D is total dominating and the induced subgraph G[V (G)−D] is a connected graph. The total outer–connected...
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Fast rns scaling of signed numbers based on the chinese remainder theorem
PublicationPrzedstawiono architekturę szybkiego skalera resztowego dla liczb ze znakiem dla modułów 5-bitowych opartego na chińskim twierdzeniu o resztach. W architekturze zastosowano wyznaczanie projekcji ortogonalnych przy użyciu funkcji logicznych 5 zmiennych, sumator kodulo m/k oraz wejściowy konwerter do systemu resztowego. Operacja modulo m/k jest wykonywana przy zastosowaniu drzewa 4-operandowych sumatorów modulo 2m/k i końcowego...
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On the ratio between 2-domination and total outer-independent domination numbers of trees
PublicationA 2-dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)D has a at least two neighbors in D. A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V(G)D is independent. The 2-domination (total outer-independent domination, respectively) number of a graph G is the minimum cardinality of a 2-dominating (total...
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Numbers, Please: Power- and Voltage-Related Indices in Control of a Turbine-Generator Set
PublicationThis paper discusses the proper selection and interpretation of aggregated control performance indices values mirroring the quality of electrical energy generation by a turbine-generator set cooperating with a power system. Typically, a set of basic/classical and individual indices is used in energy engineering to ensure the mirroring feature and is related to voltage, frequency and active or reactive power deviations from their...
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Pipelined sceling of signed residue numbers with the mixed-radix conversion in the programmable gate array
PublicationIn this work a scaling technique of signed residue numbers is proposed. The method is based on conversion to the Mixed-Radix System (MRS) adapted for the FPGA implementation. The scaling factor is assumed to be a moduli product from the Residue Number System (RNS) base. Scaling is performed by scaling of terms of the mixed-radix expansion, generation of residue reprezentation of scaled terms, binary addition of these representations...
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DO WE NEED NAVIER NUMBER? – FURTHER REMARKS AND COMPARISON WITH ANOTHER DIMENSIONLESS NUMBERS
PublicationThis paper presents a role of the Navier number (Na-dimensionless slip-length) in universal modelling of flow reported in micro- and nano-channels like: capillary biological flows, fuel cell systems, micro-electro-mechanical systems and nano-electro-mechanical systems. Similar to another bulk-like and surface-like dimensionless numbers, the Na number should be treated as a ratio of internal viscous to external viscous momentum...
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Improved magnitude estimation of complex numbers using alpha max and beta min algorithm
PublicationThe paper presents an improved algorithm for calculating the magnitude of complex numbers. This problem, which is a special case of square rooting, occurs for example, in FFT processors and complex FIR filters. The proposed method of magnitude calculation makes use of the modified alpha max and beta min algorithm. The improved version of the algorithm allows to control the maximum magnitude approximation error by using an adequate...
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Tax preferences in CIT in numbers 2009-2015
Open Research DataThese data contain information prepared by the Ministry of Finance on the value of tax preferences by areas of support in Corporate Income Tax (CIT) between 2009-2015.
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Tax preferences in PIT in numbers 2009-2015
Open Research DataThe follwoing data contain information prepared by the Ministry of Finance on the value of tax preferences by areas of support in Personal Income TAX (PIT) between 2009-2015.
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Scaling of signed residue numbers with mixed-radix conversion in FPGA with extended scaling factor selection
PublicationA scaling technique of signed residue numbers in FPGA is proposed. The technique is based on conversion of residue numbers to the Mixed-Radix System (MRS). The scaling factor is assumed to be a moduli product from the Residue Number System (RNS) base. Scaling is performed by scaling of MRS terms, the subsequent generation of residue representations of scaled terms, binary addition of these representations and generation of residues...
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Increased Certification of Semi-device Independent Random Numbers using Many Inputs and More Postprocessing
PublicationQuantum communication with systems of dimension larger than two provides advantages in information processing tasks. Examples include higher rates of key distribution and random number generation. The main disadvantage of using such multi-dimensional quantum systems is the increased complexity of the experimental setup. Here, we analyze a not-so-obvious problem: the relation between randomness certification and computational requirements...
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Implementation of magnitude calculation of complex numbers using improved alpha max plus beta min algorithm
PublicationThe paper presents the hardware implementation of the improved alpha max plus beta min algorithm for calculating the magnitude of complex numbers. This version of the algorithm requires the general division which is performed using a noniterative multiplicative division algorithm. We analyze in detail the division algorithm, its error and the impact of finite word-length signal representations on the assumed total computation error....
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Solvation numbers of manganese(II) and zinc(II) perchlorates in methanol obtained from volumetric and compressibility properties
PublicationZmierzono gęstości roztworów nadchloranów manganu(II) i cynku(II) w metanolu w temperaturach: 283.15, 288.15, 293.15, 298.15, 303.15, 308.15, 313.15 K. Ponadto wyznaczono szybkości rozchodzenia się dźwięku w badanych roztworach w 298.15 K. Uzyskane wielkości pozwoliły na obliczenie pozornych objętości molowych oraz pozornych molowych ściśliwości adiabatycznych elektrolitów w metanolu. Oszacowane wielkości graniczne wykorzystane...
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Pipelined division of signed numbers with the use of residue arithmetic for small number range with the programmable gate array
PublicationIn this work an architecture of the pipelined signed residue divider for the small number range is presented. Its operation is based on reciprocal calculation and multiplication by the dividend. The divisor in the signed binary form is used to compute the approximated reciprocal in the residue form by the table look-up. In order to limit the look-up table address an algorithm based on segmentation of the divisor into two segments...
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Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers
PublicationLet f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math....
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Database of the estimations of the numbers of simplices of triangulation of some classical Lie groups
Open Research DataIt is know that any smooth manifold can be triangulated. The number of simplices of triangulation of a given manifold depends on its topological and combinatorial structure. The data consists of the lower bounds for the numbers of simplices of each dimension of any triangulation of classical Lie groups U(n), SU(n), Sp(n), and SO(n) for n up to 25. Each...
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NUMERICAL ANALYSIS OF SPECIES DIFFUSION AND METHANOL DECOMPOSITION IN THERMOCATALYTIC REACTOR BASED ON THE INTERMETALLIC PHASE OF Ni3Al FOR LOW REYNOLDS NUMBERS
PublicationNumerical modelling of hydrogen production by means of methanol decomposition in a thermocatalytic reactor using corrugated foil made of the Ni3Al intermetallic phase is shown in the paper. Experimental results of the flow analysis of mixtures containing helium and methanol in a thermocatalytic reactor with microchannels were used for the initial calibration of the CFD calculations (calculations based on the Computational Fluid...
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Reducing the number of periodic points in the smooth homotopy class of a self-map of a simply-connected manifold with periodic sequence of Lefschetz numbers
PublicationLet f be a smooth self-map of an m-dimensional (m >3) closed connected and simply-connected manifold such that the sequence of the Lefschetz num- bers of its iterations is periodic. For a fixed natural r we wish to minimize, in the smooth homotopy class, the number of periodic points with periods less than or equal to r. The resulting number is given by a topological invariant J[f] which is defned in combinatorial terms and is...