Search results for "Unary function"
showing 10 items of 11 documents
Synthesis, Characterization, and Catalysis of β3-[(CoIIO4)W11O31(O2)4],10- the First Keggin-Based True Heteropoly Dioxygen (Peroxo) Anion. Spectrosco…
1999
Reactions of hydrogen peroxide with several lacunary polyoxometalates of the 1:11 series, XW11O39m- (X = Co3+, Ga3+, Fe3+, Si4+, and P5+), are reported. Synthetic pathways to new polyoxotungstates incorporating dioxygen moieties (peroxo and/or superoxo) are developed. The key step involves treating lacunary precursors with H2O2 in strongly buffered aqueous solutions. Upon reaction of H2O2 with α-[Co3+W11O39],9- (a) the central tetrahedral Co3+ is reduced to Co2+ and (b) each of the four unshared oxygens surrounding the vacancy are replaced by a peroxide group, yielding salts of the tetraperoxide anion β3-[(Co2+O4)W11O31(O2)4]10- (1). These results are unequivocally established by a combinat…
On lacunary Toeplitz determinants
2014
By using Riemann--Hilbert problem based techniques, we obtain the asymptotic expansion of lacunary Toeplitz determinants $\det_N\big[ c_{\ell_a-m_b}[f] \big]$ generated by holomorhpic symbols, where $\ell_a=a$ (resp. $m_b=b$) except for a finite subset of indices $a=h_1,\dots, h_n$ (resp. $b=t_1,\dots, t_r$). In addition to the usual Szeg\"{o} asymptotics, our answer involves a determinant of size $n+r$.
On bijections vs. unary functions
1996
A set of finite structures is in Binary NP if it can be characterized by existential second order formulas in which second order quantification is over relations of arity 2. In [DLS95] subclasses of Binary NP were considered, in which the second order quantifiers range only over certain classes of relations. It was shown that many of these subclasses coincide and that all of them can be ordered in a three-level linear hierarchy, the levels of which are represented by bijections, successor relations and unary functions respectively.
Descriptive Complexity, Lower Bounds and Linear Time
1999
This paper surveys two related lines of research: Logical characterizations of (non-deterministic) linear time complexity classes, and non-expressibility results concerning sublogics of existential second-order logic. Starting from Fagin’s fundamental work there has been steady progress in both fields with the effect that the weakest logics that are used in characterizations of linear time complexity classes are closely related to the strongest logics for which inexpressibility proofs for concrete problems have been obtained. The paper sketches these developments and highlights their connections as well as the obstacles that prevent us from closing the remaining gap between both kinds of lo…
Padding and the expressive power of existential second-order logics
1998
Padding techniques are well-known from Computational Complexity Theory. Here, an analogous concept is considered in the context of existential second-order logics. Informally, a graph H is a padded version of a graph G, if H consists of an isomorphic copy of G and some isolated vertices. A set A of graphs is called weakly expressible by a formula ϕ in the presence of padding, if ϕ is able to distinguish between (sufficiently) padded versions of graphs from A and padded versions of graphs that are not in A.
Unary Languages Recognized by Two-Way One-Counter Automata
2014
A two-way deterministic finite state automaton with one counter (2D1CA) is a fundamental computational model that has been examined in many different aspects since sixties, but we know little about its power in the case of unary languages. Up to our knowledge, the only known unary nonregular languages recognized by 2D1CAs are those formed by strings having exponential length, where the exponents form some trivial unary regular language. In this paper, we present some non-trivial subsets of these languages. By using the input head as a second counter, we present simulations of two-way deterministic finite automata with linearly bounded counters and linear–space Turing machines. We also show …
A Logical Characterisation of Linear Time on Nondeterministic Turing Machines
1999
The paper gives a logical characterisation of the class NTIME(n) of problems that can be solved on a nondeterministic Turing machine in linear time. It is shown that a set L of strings is in this class if and only if there is a formula of the form ∃f1..∃fk∃R1..∃Rm∀xφv; that is true exactly for all strings in L. In this formula the fi are unary function symbols, the Ri are unary relation symbols and φv; is a quantifierfree formula. Furthermore, the quantification of functions is restricted to non-crossing, decreasing functions and in φv; no equations in which different functions occur are allowed. There are a number of variations of this statement, e.g., it holds also for k = 3. From these r…
Comparative study of the arterial and lacunary systems of the ventricular myocardium of elasmobranch and teleost fishes.
1983
The typical fish heart has a spongy trabeculated ventricular myocardium (spongiosa) supplied by the venous blood of the intertrabecular spaces (lacunae); hence it is called a "venous heart." However, in some fishes a more complex ventricular muscle is found (mixed type), in which the spongiosa is covered by an outer layer of densely arranged myocardial bundles (compacta). The compacta receives oxygenated blood from the coronary vessels. The objective of this study was to investigate relations between myoarchitecture and blood supply with an emphasis on the hitherto unexplored, putative vascular connections between the arterial and the lacunary circuits. Using histological methods combined w…
Synthesis and Characterization of New Keggin-Based Structure Tungstocobaltates: K9[Co3+W11O39] and its Vanadium Derivatives, K8[Co2+W11V4+O40], K7[Co…
1995
Abstract Good yield synthesis of the potassium salts of four new interrelated : olyoxometalates, [Co3+W11O39]9-(I), [Co2+W11V4+O4]8-(II), [Co2+W11V5+O40]7-(III), and [Co3+W11V5+O40]6-(IV), are reported. The title compounds have been characterized by IR and UV-Vis.-NIR spectroscopies, cyclic voltammetry and magnetic measurements. IR spectra confirm that I has the lacunary structure of an undecatungstate anion, while II through IV have a substituted α-Keggin structure with the vanadium atom located at an octahedral site replacing one tungsten atom. Optical data confirm the oxidation states for both, the Co and V atoms, as formulated for I through IV.
Lacunary Bifurcation of Multiple Solutions of Nonlinear Eigenvalue Problems
1991
In order to describe the type of nonlinear eigenvalue problems we are going to discuss, consider a densely defined closed linear operator T in a real Hilbert space H and let H1 be the Hilbert space which consists of the domain of T together with the graph norm. Also, let H 1 * be the dual space of H1 and denote the dual operator corresponding to T: H1 → H by T’:H → H 1 * . Since H1 is dense in H, we may view H as a subspace of H1, and then the scalar product (·,·) on H and the dual pairing on H1 × H 1 * coincide on H1 × H.