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Abstract A ring R is said to be of stable range 1.5 if for each a , b ∈ R and 0 ≠ c ∈ R satisfying a R + b R + c R = R there exists r ∈ R such that (a + b r) R + c R = R. Let R be a commutative domain in which all finitely generated ideals are principal, and let R be of stable range 1.5. Then each matrix A over R is reduced to Smith's canonical form by transformations PAQ in which P and Q are invertible and at least one of them can be chosen to be a product of elementary matrices. We generalize Helmer's theorem about the greatest common divisor of entries of A over R. [ABSTRACT FROM AUTHOR]

The quaternion linear canonical transforms (QLCT) is a nontrivial generalization of the linear canonical transform (LCT) using quaternion algebra. Due to the noncommutative property of quaternion multiplication there are different definitions for the QLCT. We establish the relationships among different types of the QLCTs. [ABSTRACT FROM AUTHOR]

In this paper, we show that a parallel differential form Ψ of even degree on a Riemannian manifold allows to define a natural differential both on Ω ∗ (M) and Ω ∗ (M , T M) , defined via the Frölicher–Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential with respect to the canonical parallel 4 -form on a G 2 - and Spin (7) -manifold, respectively. We calculate the cohomology groups of Ω ∗ (M) and give a partial description of the cohomology of Ω ∗ (M , T M). [ABSTRACT FROM AUTHOR]

We discuss Gauss codes of virtual diagrams and virtual doodles. The notion of a left canonical Gauss code is introduced and it is shown that oriented virtual doodles are uniquely presented by left canonical Gauss codes. [ABSTRACT FROM AUTHOR]

Given a rank n irreducible finite reflection group W , the W -invariant polynomial functions defined in R n can be written as polynomials of n algebraically independent homogeneous polynomial functions, p 1 ( x ) , … , p n ( x ) , called basic invariant polynomials. Their degrees are well known and typical of the given group W . The polynomial p 1 ( x ) has the lowest degree, equal to 2. It has been proved that it is possible to choose all the other n − 1 basic invariant polynomials in such a way that they satisfy a certain system of differential equations, including the Laplace equations △ p a ( x ) = 0 , a = 2 , … , n , and so are harmonic functions. Bases of this kind are called canonical. Explicit formulas for canonical bases of invariant polynomials have been found for all irreducible finite reflection groups, except for those of types E 6 , E 7 and E 8 . Those for the groups of types E 6 , E 7 and E 8 are determined in this article. [ABSTRACT FROM AUTHOR]

Variational and divergence symmetries are studied in this paper for linear equations of maximal symmetry in canonical form, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. Some of these results apply to linear equations of a general form and of arbitrary orders or having a symmetry algebra of arbitrary dimension. [ABSTRACT FROM AUTHOR]

Let ( X , h ) be a compact and irreducible Hermitian complex space of complex dimension m . In this paper we are interested in the Dolbeault operator acting on the space of L 2 sections of the canonical bundle of reg ( X ) , the regular part of X . More precisely let d ‾ m , 0 : L 2 Ω m , 0 ( reg ( X ) , h ) → L 2 Ω m , 1 ( reg ( X ) , h ) be an arbitrarily fixed closed extension of ∂ ‾ m , 0 : L 2 Ω m , 0 ( reg ( X ) , h ) → L 2 Ω m , 1 ( reg ( X ) , h ) where the domain of the latter operator is Ω c m , 0 ( reg ( X ) ) . We establish various properties such as closed range of d ‾ m , 0 , compactness of the inclusion D ( d ‾ m , 0 ) ↪ L 2 Ω m , 0 ( reg ( X ) , h ) where D ( d ‾ m , 0 ) , the domain of d ‾ m , 0 , is endowed with the corresponding graph norm, and discreteness of the spectrum of the associated Hodge–Kodaira Laplacian d ‾ m , 0 ⁎ ∘ d ‾ m , 0 with an estimate for the growth of its eigenvalues. Several corollaries such as trace class property for the heat operator associated to d ‾ m , 0 ⁎ ∘ d ‾ m , 0 , with an estimate for its trace, are derived. Finally in the last part we provide several applications to the Hodge–Kodaira Laplacian in the setting of both compact irreducible Hermitian complex spaces with isolated singularities and complex projective surfaces. [ABSTRACT FROM AUTHOR]

A series of 25 Greek coins from the 6th to 4th centuries BC was studied by PIXE for their trace element composition, with an aim to discover the origin of their silver ore. The procedure revealed a counterfeited coin, and then concentrated on distinguishing the coins minted from the ore of Laurion on the Attica peninsula and the coins minted from other sources. Linear discriminant analysis based on the impurities and alloying elements of copper, gold, lead and bismuth revealed that discrimination is indeed possible according to a single canonical variable. [ABSTRACT FROM AUTHOR]

We work on the analytical solution of the stochastic differential equations (SDE) via invariant approaches. In particularly, we focus on the stochastic Black-Derman Toy (BDT) interest rate model, among others. After we present corresponding (1+1) parabolic linear PDE for BDT-SDE, we use theoretical framework about the invariant approaches for the (1+1) linear PDE being done in the literature. We show that it is not possible to reduce BDT-PDE into the first and second Lie canonical forms. On the other hand, we success to find transformations for reducing it to the third Lie canonical form. After that, we obtain analytical solution of BDT-PDE by using these transformations. Moreover, we conclude that it can be reduced to the fourth Lie canonical form but, to the best of our knowledge, its analytical solution in this form is hard to find yet. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. The zero-divisor graph of R is the (simple) graph Г(R) with vertices the nonzero zero-divisors of R, and two distinct vertices x and y are adjacent if and only if xy = 0. In this article, we investigate Г(R) when R does not have an identity, and we determine all such zero-divisor graphs with 14 or fewer vertices. [ABSTRACT FROM AUTHOR]

It is well-known that in general polynomials lose their CNS property by addition of small positive integers. We comment on a conjecture of S. Akiyama on addition of sufficiently large positive constants to CNS polynomials. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce new invariants of Cohen–Macaulay local rings. Our focus is the class of Cohen–Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C , there are integers–the type of R , the reduction number of C –that provide valuable metrics to express the deviation of R from being a Gorenstein ring. We enlarge this list with other integers–the roots of R and several canonical degrees. The latter are multiplicity based functions of the Rees algebra of C . [ABSTRACT FROM AUTHOR]

We give a characterization of complete strongly dually atomic lattices having unique irredundant decompositions which are also canonical. It is shown that all known characterizations of lattices with unique irredundant decompositions are a consequence of this result. In addition, upper continuous closure lattices of convex geometries with (unique) irredundant decompositions are characterized. [ABSTRACT FROM AUTHOR]

We give an algorithm which requires no integer factorization for computing the canonical height of a point in ℙ1(ℚ) relative to a morphism φ: ℙ1ℚ→ ℙ1 ℚℚ of degree d ≥ 2. [ABSTRACT FROM AUTHOR]

This paper proposes a canonical variate analysis (CVA) approach based on feature representation of canonical correlation for the monitoring of faults associated with changes in process correlations, which involves two new metrics, R s and R r , corresponding to the state and residual spaces. The utilization of the canonical correlation feature can improve the monitoring proficiency by providing more application-dependent representations compared with the original data, as well as a decreased degree of redundancy in the feature space. A physical interpretation is provided for the canonical correlation-based method. The effectiveness of the proposed approach for the monitoring of process correlation changes is demonstrated for both abrupt (step change) and incipient (slow drift) types of faults in simulation studies of a network system. In the simulation results, the canonical correlation-based method has superior performance over both the causal dependency-based method and the traditional variable-based method. [ABSTRACT FROM AUTHOR]

Differential invariants and their corresponding canonical forms for systems of three 2nd-order ODEs possessing three-dimensional Lie algebras are constructed. Their extension up to kth-order system of three 2nd-order ODEs is presented. Furthermore singularity in invariant structure for the canonical forms is investigated. In addition integrability of these canonical forms is discussed. Illustrative physical examples from mechanics of system of particles are provided. [ABSTRACT FROM AUTHOR]

In this paper we point out the relationship between linear MRD-codes and various geometric objects as linearized polynomials, linear sets of P G ( n − 1 , q n ) , and generalized Segre varieties. We introduce and characterize a class of particular linear MRD-codes called F q n -linear codes with distance n − k + 1 proving that such codes are equivalent to ( n − k + 1 ) -embeddings of a canonical subgeometry. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to examine stochastic Markovian models for circuits in phase space for which the drift term is equivalent to the standard circuit equations. In particular, we include dissipative components corresponding to both a resistor and a memristor in series. We obtain a dilation of the problem which is canonical in the sense that the underlying Poisson bracket structure is preserved under the stochastic flow. We do this first of all for standardWiener noise but also treat the problem using a new concept of symplectic noise, where the Poisson structure is extended to the noise as well as the circuit variables, and in particular where we have canonically conjugate noises. Finally, we construct a dilation which describes the quantum mechanical analogue. [ABSTRACT FROM AUTHOR]

We present necessary and sufficient conditions with explicit frame bounds for a discrete wavelet system of the form { D a T k ϕ } a ∈ U ( N ) , k ∈ I N to be a frame for the unitary space C N . It is shown that the canonical dual of a discrete wavelet frame for C N has the same structure. This is not true (well known) for canonical dual of a wavelet frame for L 2 ( R ) . Several numerical examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

Retinal amacrine cells express nitric oxide (NO) synthase and produce NO, making NO available to regulate the function of amacrine cells. Here we test the hypothesis that NO can alter the GABAergic synaptic output of amacrine cells. We investigate this using whole cell voltage clamp recordings and Ca2+ imaging of cultured chick retinal amacrine cells. When recording from amacrine cells receiving synaptic input from other amacrine cells, we find that NO increases GABAergic spontaneous postsynaptic current (sPSC) frequency. This increase in sPSC frequency does not require the canonical NO receptor, soluble guanylate cyclase, or presynaptic action potentials. However, removal of extracellular Ca2+ and buffering of cytosolic Ca2+ both inhibit the response to NO. In Ca2+ imaging experiments, we confirm that NO increases cytosolic Ca2+ in amacrine cell processes by activating a Ca2+ influx pathway. Neither the increase in sPSC frequency nor the cytosolic Ca2+ elevations are dependent upon Ca2+ release from stores. NO also enhances evoked GABAergic responses. Because voltage-gated Ca2+ channel function is not altered by NO, the increased evoked response is likely due to the combined effect of voltage-dependent Ca2+ influx adding to the NO-dependent, voltage-independent, Ca2+ influx. Insight into the identity of the Ca2+ influx pathway is provided by the transient receptor potential canonical (TRPC) channel inhibitor clemizole, which prevents the NO-dependent increase in sPSC frequency and cytosolic Ca2+ elevations. These data suggest that NO production in the inner retina will enhance Ca2+ -dependent GABA release from amacrine cells by activating TRPC channel(s). [ABSTRACT FROM AUTHOR]

In this paper a detailed Hamiltonian analysis of three-dimensional gravity without dynamics proposed by V. Hussain is performed. We report the complete structure of the constraints and the Dirac brackets are explicitly computed. In addition, the Faddeev–Jackiw symplectic approach is developed; we report the complete set of Faddeev–Jackiw constraints and the generalized brackets, then we show that the Dirac and the generalized Faddeev–Jackiw brackets coincide to each other. Finally, the similarities and advantages between Faddeev–Jackiw and Dirac’s formalism are briefly discussed. [ABSTRACT FROM AUTHOR]

An advanced Gauss pseudospectral method (AGPM) was proposed to estimate the parameters of the continuous-time (CT) Hammerstein model. The nonlinear part of the Hammerstein system is approximated with pseudospectral approximation method. The linear part was written as a controllable canonical form to circumvent the high order time-derivative of the input and output (I/O) signals, which could multiply the measurement noise in the identification procesion. Furthermore, an output error minimization was constructed for the CT Hammerstein model identification, which was then transcribed into a nonlinear programming (NLP) problem by AGPM. AGPM could converge to the true values of the CT Hammerstein model wiii few interpolated Legendre-Gauss (LG) nodes. Lastly, two illustrative examples were proposed to verify the accuracy and efficiency of the method. [ABSTRACT FROM AUTHOR]

Low-rank tensor completion addresses the task of filling in missing entries in multi-dimensional data. It has proven its versatility in numerous applications, including context-aware recommender systems and multivariate function learning. To handle large-scale datasets and applications that feature high dimensions, the development of distributed algorithms is central. In this work, we propose novel, highly scalable algorithms based on a combination of the canonical polyadic (CP) tensor format with block coordinate descent methods. Although similar algorithms have been proposed for the matrix case, the case of higher dimensions gives rise to a number of new challenges and requires a different paradigm for data distribution. The convergence of our algorithms is analyzed and numerical experiments illustrate their performance on distributed-memory architectures for tensors from a range of different applications. [ABSTRACT FROM AUTHOR]

Boney and Grossberg [7] proved that every nice AEC has an independence relation. We prove that this relation is unique: in any given AEC, there can exist at most one independence relation that satisfies existence, extension, uniqueness and local character. While doing this, we study more generally the properties of independence relations for AECs and also prove a canonicity result for Shelah's good frames. The usual tools of first-order logic (like the finite equivalence relation theorem or the type amalgamation theorem in simple theories) are not available in this context. In addition to the loss of the compactness theorem, we have the added difficulty of not being able to assume that types are sets of formulas. We work axiomatically and develop new tools to understand this general framework. [ABSTRACT FROM AUTHOR]

In this paper, we show that the secant variety to a smooth projective variety embedded by a sufficiently positive line bundle is normal. As an application, we deduce that the secant variety to a general canonical curve of genus at least 7 is normal. [ABSTRACT FROM AUTHOR]

We present the construction of the Maslov canonical operator adapted to an arbitrary coordinate system on the corresponding Lagrangian manifold. The construction does not require any additional choice of the phase function. [ABSTRACT FROM AUTHOR]

We prove that the canonical decompositions of hyperbolic fibered two-bridge link complements are layered triangulations, by showing that A'Campo's criterion for detecting fiberedness works for all hyperbolic fibered two-bridge links. [ABSTRACT FROM AUTHOR]

Let T 0 be the once-punctured torus and θ a real number with 0 < θ < 2 π . Let ρ ˜ be a representation of π 1 ( T 0 ) to SL ( 2 , R ) which sends the peripheral loop to an elliptic element with trace − 2 cos ⁡ ( θ / 2 ) . Let ρ be the PSL ( 2 , R ) -representation induced from ρ ˜ , and assume it satisfies Bowditch's Q-condition. In this paper, we construct a certain polyhedron, which is obtained as a variation of Jorgensen's theory to cone manifolds, and construct a complete cone hyperbolic structure on the 3-dimensional cone manifold obtained as the product of the torus with a single cone point and R which induces ρ as the holonomy representation. [ABSTRACT FROM AUTHOR]

We define a reasonably well-behaved class of ultraimaginaries, i.e. classes modulo -invariant equivalence relations, called tame, and establish some basic simplicity-theoretic facts. We also show feeble elimination of supersimple ultraimaginaries: If is an ultraimaginary definable over a tuple with , then is eliminable up to rank . Finally, we prove some uniform versions of the weak canonical base property. [ABSTRACT FROM AUTHOR]

We study invariant types in NIP theories. Amongst other things: we prove a definable version of the -theorem in theories of small or medium directionality; we construct a canonical retraction from the space of -invariant types to that of -finitely satisfiable types; we show some amalgamation results for invariant types and list a number of open questions. [ABSTRACT FROM AUTHOR]

The conformal loop ensemble $${{\mathrm{CLE}}}_\kappa $$ with parameter $$8/3 < \kappa < 8$$ is the canonical conformally invariant measure on countably infinite collections of non-crossing loops in a simply connected domain. We show that the number of loops surrounding an $$\varepsilon $$ -ball (a random function of $$z$$ and $$\varepsilon $$ ) minus its expectation converges almost surely as $$\varepsilon \rightarrow 0$$ to a random conformally invariant limit in the space of distributions, which we call the nesting field. We generalize this result by assigning i.i.d. weights to the loops, and we treat an alternate notion of convergence to the nesting field in the case where the weight distribution has mean zero. We also establish estimates for moments of the number of CLE loops surrounding two given points. [ABSTRACT FROM AUTHOR]

Let p be a positive integer and let A ( p ) ( R , S ) be the class of nonnegative integral matrices with entries less than or equal to p , with row–sum partition R , and column–sum partition S . In this paper we state a new necessary and sufficient condition for A ( p ) ( R , S ) ≠ ∅ . This condition generalizes the well known Gale–Ryser theorem. We also present a canonical construction for matrices in A ( p ) ( R , S ) . [ABSTRACT FROM AUTHOR]

We study the holomorphic equivalence problem for finite type hypersurfaces in C 2 . We discover a geometric condition, which is sufficient for the existence of a natural convergent normal form for a finite type hypersurface. We also provide an explicit construction of such a normal form. As an application, we construct a canonical connection for a large class of finite type hypersurfaces. To the best of our knowledge, this gives the first construction of an invariant connection for Levi-degenerate hypersurfaces in C 2 . [ABSTRACT FROM AUTHOR]

Employing smart Monte Carlo sampling techniques within the grand canonical ensemble, we investigate the properties of water at a model hydrophobic substrate. By reducing the strength of substrate-water attraction, we find that fluctuations in the local number density, quantified by a rigorous definition of the local compressibility X(z), increase rapidly for distances z within one or two molecular diameters from the substrate as the degree of hydrophobicity, measured by the macroscopic contact angle 9, increases. Our simulations provide evidence for a continuous (critical) drying transition as the substrate-water interaction becomes very weak: cos(θ) → -1 . We speculate that the existence of such a transition might account for earlier simulation observations of strongly enhanced density fluctuations. [ABSTRACT FROM AUTHOR]

Correlating Shannon's maximum informational entropy variational principle with the constant value of Onicescu's informational energy, the uncertainty relations for canonical systems with SL(2R) invariance are obtained. The constant value of Onicescu's informational energy corresponds, through transitivity manifolds of the SL(2R) group, to the ergodic theorem and, in particular case of a linear oscillator, to a quantification condition. Specifying de Broglie's idea by a periodic field in a complex coordinate, it is proved that the oscillator synchronization group of the same ensemble is still an achievement of the SL(2R) (Barbilian's group). Integrally invoking the invariant functions through the simultaneous action of the two isomorphic SL(2R) groups, the uncertainty relations and Stoler group of synchronization among oscillators from different ensembles (i.e., the second quantification when the annihilation and creation operators refer to a linear oscillator) are obtained. Barbilian's group parameters are interpreted by means of a variational principle as field amplitudes of a stationary vacuum metric. [ABSTRACT FROM AUTHOR]

We study the structure of the canonical Poincaré-Lindstedt perturbation series in the Deprit operator formalism and establish its connection to the Kato resolvent expansion. A discussion of invariant definitions for averaging and integrating perturbation operators and their canonical identities reveals a regular pattern in the series for the Deprit generator. This regularity is explained using Kato series and the relation of the perturbation operators to the Laurent coefficients for the resolvent of the Liouville operator. This purely canonical approach systematizes the series and leads to an explicit expression for the Deprit generator in any order of the perturbation theory: $G = - \hat S_H H_j $, where $\hat S_H $ is the partial pseudoinverse of the perturbed Liouville operator. The corresponding Kato series provides a reasonably effective computational algorithm. The canonical connection of the perturbed and unperturbed averaging operators allows describing ambiguities in the generator and transformed Hamiltonian, while Gustavson integrals turn out to be insensitive to the normalization style. We use nonperturbative examples for illustration. [ABSTRACT FROM AUTHOR]

In this article, we compute the weighted log canonical thresholds of toric plurisubharmonic functions, i.e. convex increasing functions of the logarithms of the absolute values of their complex arguments. [ABSTRACT FROM AUTHOR]

We analyze the efficacy of a standard manifold-based reduction method used to simplify reaction dynamics and find conditions under which the reduction can succeed and fail. In the standard reduction, a heteroclinic trajectory linking saddle and sink equilibria is taken as a candidate reduced manifold which we call a Canonical Invariant Manifold (CIM). We develop and exercise analytic tools for studying the local behavior of trajectories near the CIM. In so doing, we find conditions under which nearby trajectories are attracted to the CIM (ACIM) as well as conditions for which the dynamics on the ACIM are slow (SACIM). The method is demonstrated on a (1) simple model problem, (2) Zel'dovich mechanism for nitric oxide production, and (3) hydrogen-air system. For systems that evolve in a three-dimensional composition space, we find that normal stretching away from the CIM in a volume-shrinking vector field is admitted and that depending on the magnitude of the local rotation rate, may or may not render the CIM to be attractive. The success and failure of the candidate CIM as a SACIM is displayed for the model system. Results for the Zel'dovich mechanism and hydrogen-air systems are less definitive, though for specific conditions a SACIM is identified for both systems. [ABSTRACT FROM AUTHOR]

The purpose of this paper and its sequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature, as well as classical toric varieties. In this paper, we define a toric stack as the stack quotient of a toric variety by a subgroup of its torus (we also define a generically stacky version). Any toric stack arises from a combinatorial gadget called a stacky fan. We develop a dictionary between the combinatorics of stacky fans and the geometry of toric stacks, stressing stacky phenomena such as canonical stacks and good moduli space morphisms. We also show that smooth toric stacks carry a moduli interpretation extending the usual moduli interpretations of Pn and [A¹/Gm]. Indeed, smooth toric stacks precisely solve moduli problems specified by (generalized) effective Cartier divisors with given linear relations and given intersection relations. Smooth toric stacks therefore form a natural closure to the class of moduli problems introduced for smooth toric varieties and smooth toric DM stacks in papers by Cox and Perroni, respectively. We include a plethora of examples to illustrate the general theory. We hope that this theory of toric stacks can serve as a companion to an introduction to stacks, in much the same way that toric varieties can serve as a companion to an introduction to schemes. [ABSTRACT FROM AUTHOR]

Let k be a field of characteristic 0, and let (F,R) be a Landweber exact formal group law. We consider a Landweber exact T-spectrum ε := R∅L MGL and its effective cover f0ε → ε with respect to Voevodsky’s slice tower. The coefficient ring R0 of f0ε is the subring of R consisting of elements of R of nonpositive degree; the power series F ∈ R[[u, v]] has coefficients in R0, although (F,R0) is not necessarily Landweber exact. We show that the geometric part X → f0 ε∗(X) := (f0 ε)2∗,∗(X) of f0ε is canonically isomorphic to the oriented cohomology theory X → R0∅L Ω∗(X), where Ω∗ is the theory of algebraic cobordism as defined in [12]. This recovers results of Dai–Levine [2] as the special case of algebraic K-theory and its effective cover, connective algebraic K-theory. [ABSTRACT FROM AUTHOR]

We prove birational boundedness results on complete intersections with trivial canonical class of base point free divisors in (some version of) Fano varieties. Our results imply in particular that Batyrev–Borisov toric construction produces only a bounded set of Hodge numbers in any given dimension, even as the codimension is allowed to grow. [ABSTRACT FROM AUTHOR]

We study the homological behavior ofmodules over local rings modulo exact zero-divisors.We obtain new results which are in some sense "opposite" to those known for modules over local rings modulo regular elements. [ABSTRACT FROM AUTHOR]

Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier-Sato-Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra. In the second part of the paper, we introduce new quantizations and show that all quantum cluster monomials in our setting belong to the dual canonical basis of the corresponding quantum unipotent subgroup. This result generalizes previous work by Lampe and by Hernandez-Leclerc from the Kronecker and Dynkin quiver case to the acyclic case. The Fourier transform part of this paper provides crucial input for the second author's paper where he constructs bases of acyclic quantum cluster algebras with arbitrary coefficients and quantization. [ABSTRACT FROM AUTHOR]

For three-dimensional Schrödinger equations, we study how to localize exact solutions represented as the product of an Airy function (Berry-Balazs solutions) and a Bessel function and known as Airy-Bessel beams in the paraxial approximation in optics. For this, we represent such solutions in the form of Maslov's canonical operator acting on compactly supported functions on special Lagrangian manifolds. We then use a result due to Hörmander, which permits using the formula for the commutation of a pseudodifferential operator with Maslov's canonical operator to 'move' the compactly supported amplitudes outside the canonical operator and thus obtain effective formulas preserving the structure based on the Airy and Bessel functions. We discuss the influence of dispersion effects on the obtained solutions. [ABSTRACT FROM AUTHOR]

Abstract: We will present an elementary algebraic construction of canonical bases for modified quantum and q-Schur algebras. Furthermore, we will discuss the relation between the canonical basis of the positive part of quantum and the canonical basis of modified quantum . Finally, we will establish an explicit connection between the canonical basis of modified quantum and that of q-Schur algebras. [Copyright &y& Elsevier]

Abstract: A canonical system of basic invariants is a system of invariants satisfying a set of differential equations. The properties of a canonical system are related to the mean value property for polytopes. In this article, we naturally identify the vector space spanned by a canonical system of basic invariants with an invariant space determined by a fundamental antiinvariant. From this identification, we obtain explicit formulas of canonical systems of basic invariants. The construction of the formulas does not depend on the classification of finite irreducible reflection groups. [Copyright &y& Elsevier]

The two main objectives of this paper are (a) to prove purely topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. In previously known dualities for semilattices and bounded lattices, the dual spaces are compact 0-dimensional spaces with additional algebraic structure. For example, semilattices are dual to 0-dimensional compact semilattices. Here we establish dual categories in which the spaces are characterized purely in topological terms, with no additional algebraic structure. Thus the results can be seen as generalizing Stone's duality for distributive lattices rather than Priestley's. The paper is the first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by canonical systems, is constant on C. This provides an alternative proof of the De Branges theorem that the canonical systems with trHl imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a canonical system. [ABSTRACT FROM AUTHOR]

Using horizontal n-bases of the tangent bundle of the linear frame bundle FM of an n-dimensional manifold M, the canonical form in the non-holonomic second-order frame bundle of M is introduced as a restriction of the canonical form of the bundle TFM → FM. This construction generalizes the ones in the corresponding semi-holonomic and holonomic second-order frame bundles. We prove that the natural projection of the set of all non-holonomic second-order frames of M into FM defines a principal bundle structure. [ABSTRACT FROM AUTHOR]