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One important algebraic invariant in networks is complexity. This invariant ensures the accuracy and dependability of the network. In this paper, we employ a combinatorial approach to determine the graph's complexity. A fundamental set of building blocks (basic graphs) will serve as the foundation for all the graphs we investigate, after which we will analyze the individual blocks and the ways in which they are connected. We compute the spectrum and complexity of a number of fundamental graphs and then we employ the novel duplication corona and Cartesian product operations to construct advanced networks from these graphs. Specifically, straightforward formulas are derived for the complexity of the networks created by the new duplicating corona of the regular graphs (prism, diagonal prism, cycle, complete graph, shadow of the cycle, and Petersen graph) with some families of graphs. Furthermore, using Cartesian product operation, evident and specific formulas for the complexity of the prism of the grid graph Gl,κ, the prism of the stacked book graph Bl,s, the diagonal plane prism grid graph, and the prism of the cylindrical graph Cl,κ are derived. [ABSTRACT FROM AUTHOR]

In this work we study the change of the structure of a regular pencil when we perform small perturbations over some of its rows and the other rows remain unaltered. We provide necessary conditions when several rows are perturbed, and prove them to be sufficient to prescribe the homogenous invariant factors or the Weyr characteristic of the resulting pencil when one row is perturbed. [ABSTRACT FROM AUTHOR]

Computing the autotopism group of a partial Latin rectangle can be performed in a variety of ways. This pilot study has two aims: (a) to compare these methods experimentally and (b) to identify the design goals one should have in mind for developing practical software. To this end, we compare six families of algorithms (two backtracking methods and four graph automorphism methods), with and without the use of entry invariants, on two test suites. We consider two entry invariants: one determined by the frequencies of row, column, and symbol representatives, and one determined by 2×2 submatrices. We find: (a) with very few entries, many symmetries often exist, and these should be identified mathematically rather than computationally, (b) with an intermediate number of entries, a quick‐to‐compute entry invariant was effective at reducing the need for computation, (c) with an almost‐full partial Latin rectangle, more sophisticated entry invariants are needed, and (d) the performances for (full) Latin squares and almost‐full partial Latin squares are significantly poorer than other partial Latin rectangles of comparable size, obstructed by the existence of (partial) Latin squares with large (possibly transitive) autotopism groups. [ABSTRACT FROM AUTHOR]

Action Units activation is a set of local individual facial muscle parts that occur in time constituting a natural facial expression event. AUs occurrence activation detection can be inferred as temporally consecutive evolving movements of these parts. Detecting AUs automatically can provide explicit benefits since it considers both static and dynamic facial features. Our work is divided into three contributions: first, we extracted the features from Local Binary Patterns, Local Phase Quantisation, and dynamic texture descriptor LPQTOP with two distinct leveraged network models from different CNN architectures for local deep visual learning for AU image analysis. Second, cascading the LPQTOP feature vector with Long Short‐Term Memory is used for coding longer term temporal information. Next, we discovered the importance of stacking LSTM on top of CNN for learning temporal information in combining the spatially and temporally schemes simultaneously. Also, we hypothesised that using an unsupervised Slow Feature Analysis method is able to leach invariant information from dynamic textures. Third, we compared continuous scoring predictions between LPQTOP and SVM, LPQTOP with LSTM, and AlexNet. A competitive substantial performance evaluation was carried out on the Enhanced CK dataset. Overall, the results indicate that CNN is very promising and surpassed all other methods [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a computational tool that checks theoretical conditions in order to determine whether a weighted graph, as a topological invariant of stable maps, can be associated to stable maps without cusps (ie, fold maps) from closed surfaces to the projective plan. [ABSTRACT FROM AUTHOR]

In this article, we give, under the Riemann hypothesis, an upper bound for the exponential moments of the imaginary part of the logarithm of the Riemann zeta function on the critical line. Our result, which gives information on the fluctuations of the distribution of the zeros of ζ, has the same accuracy as the result obtained by Soundararajan (Ann. of Math. (2)170(2) (2009), 981–993) for the moments of |ζ|. [ABSTRACT FROM AUTHOR]

It is established that every cubic form in at least eight variables which can be realised as a diagonal form on a hyperplane has a non‐trivial p‐adic solution for all primes p. [ABSTRACT FROM AUTHOR]

We study the smoothness properties of planar curves γ⊂R2, 0∈γ, which are invariant under a local real‐analytic diffeomorphism ψ fixing the origin. Under certain conditions, depending on the first‐order jet (if the eigenvalues of dψ(0) are not both of modulus one) or on a higher order jet (if ψ is tangent to the identity) of ψ and γ, we show that γ must be real analytic as soon as it is smooth enough — in particular, if it is of class C∞. On the other hand, when these conditions are not verified we can construct examples of nowhere‐analytic curves of class C∞, whose Taylor expansion is divergent at 0, which are invariant under non‐trivial real‐analytic local diffeomorphisms (either tangent to the identity or not). [ABSTRACT FROM AUTHOR]

Electromagnetic multipoles serve as one of the most ubiquitous languages in photonics, forming a complete expansion basis for radiations of arbitrary types. Electromagnetic multipoles from the perspective of line fields and singularities are explored. It is then discovered that, for arbitrary finite sources in a homogeneous background, there have to be isolated directions along which the radiation is either zero or circularly polarized. For such singular directions, half‐integer Hopf indices can be assigned and the index sum has to be a global invariant of 2. With this topological insight, the hidden dimensions of radiative circularly polarized Bloch modes of photonic crystal slabs are unveiled, revealing their topological origins of line singularities and indices. This work has established subtle connections between three seemingly unrelated but sweeping physical entities (line singularities of Hopf indices, electromagnetic multipoles, and Bloch modes), which can nourish new frames of visions and applications fertilizing many related fields. [ABSTRACT FROM AUTHOR]

The system of generalized Chaplygin gas equations with a coulomblike friction term has been investigated by using the famous Lie symmetry method. A direct and systematic algorithm based on the adjoint transformation and invariants of the admitted Lie algebras is then used to construct one‐ and two‐dimensional optimal system of the Chaplygin gas equations. Inequivalent classes of group invariant solutions are then obtained using the one‐dimensional optimal system. Further, the evolutionary behaviour of the weak discontinuity wave within the state characterized by one of the group invariant solutions is investigated in detail, and certain observations are noted in respect to their contrasting behaviour. [ABSTRACT FROM AUTHOR]

Lie symmetries are applied to classify the source of the magnetic field for the Pulsar equation near to the surface of the neutron star. We find that there are six possible different admitted Lie algebras. We apply the corresponding Lie invariants to reduce the Pulsar equation close to the surface to an ordinary differential equation. This equation is solved either with the use of Lie symmetries or the application of the ARS algorithm for singularity analysis to write the analytic solution as a Laurent expansion. These solutions are called inner solutions. [ABSTRACT FROM AUTHOR]

In this paper, by introducing a new frame on spacelike curves lying in lightcone 3‐space, we investigate the geometric properties of the lightlike surface of the Darboux‐like indicatrix and the lightlike surface of the binormal indicatrix generated by spacelike curves in lightcone 3‐space. As an extension of our previous work and an application of the singularity theory, the singularities of the lightlike surfaces of the Darboux‐like indicatrix and the lightlike surface of the binormal indicatrix are classified, several new invariants of spacelike curves are discovered to be useful for characterizing these singularities, meanwhile, it is found that the new invariants also measure the order of contact between spacelike curves or principal normal indicatrixes of spacelike curves located in lightcone 3‐space and two‐dimensional lightcone whose vertices are at the singularities of lightlike surfaces. One concrete example is provided to illustrate our results. [ABSTRACT FROM AUTHOR]

By combining analytic and geometric viewpoints on the concentration of the curvature of the Milnor fibre, we prove that Lipschitz homeomorphisms preserve the zones of multi‐scale curvature concentration as well as the gradient canyon structure of holomorphic functions of two variables. This yields the first new Lipschitz invariants after those discovered by Henry and Parusiński in 2003. [ABSTRACT FROM AUTHOR]

We adapt Grayson's model of higher algebraic K‐theory using binary acyclic complexes to the setting of stable ∞‐categories. As an application, we prove that the K‐theory of stable ∞‐categories preserves infinite products. [ABSTRACT FROM AUTHOR]

The Seiberg-Witten equation with multiple spinors generalises the classical Seiberg-Witten equation in dimension 3. In contrast to the classical case, the moduli space of solutions M can be non-compact due to the appearance of so-called Fueter sections. In the absence of Fueter sections we define a signed count of points in M and show its invariance under small perturbations. We then study the equation on the product of a Riemann surface and a circle, describing Min terms of holomorphic data over the surface. We define analytic and algebro-geometric compactifications of M, and construct a homeomorphism between them. For a generic choice of circle-invariant parameters of the equation, Fueter sections do not appear andMis a compact Kähler manifold. After a perturbation it splits into isolated points which can be counted with signs, yielding a number independent of the initial choice of the parameters. We compute this number for surfaces of low genus. [ABSTRACT FROM AUTHOR]

We construct cobordism maps on link Floer homology associated to decorated link cobordisms. The maps are defined on a curved chain homotopy type invariant. We describe the construction and prove invariance. We also make a comparison with the graph TQFT for Heegaard Floer homology. [ABSTRACT FROM AUTHOR]

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4‐manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fundamental form is suitably bounded. [ABSTRACT FROM AUTHOR]

It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form Gμ(α,β)=C1(μ(M))∫Mαμβμμ+C2(μ(M))∫Mα·∫Mβfor some smooth functions C1,C2 of the total volume μ(M). Here we determine the geodesics and the curvature of this metric and study geodesic and metric completeness. [ABSTRACT FROM AUTHOR]

A curious result from mixed linear models applied to genome‐wide association studies was expanded. In particular, a model in which one or more markers are considered as fixed but are allowed to contribute to the covariance structure by treating such markers as random as well was examined. The best linear unbiased estimator of marker effects is invariant with respect to whether those markers are employed in constructing a genomic relationship matrix or are ignored, provided marker effects are uncorrelated with those not being tested. Also, the implications of regarding some marker effects as fixed when, in fact, these possess a non‐trivial covariance structure with those declared as random were examined. [ABSTRACT FROM AUTHOR]

We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree. [ABSTRACT FROM AUTHOR]

From the nonlocal symmetries of the Whitham-Broer-Kaup system, an eight-dimensional Lie algebra is found and the corresponding one-dimensional optimal system is constructed to provide an inequivalent classification. Six types of inequivalent group invariant solutions are demonstrated, some of which reflect the interactions between soliton and other nonlinear waves. [ABSTRACT FROM AUTHOR]

Relative geometric invariant theory studies the behavior of semistable points under equivariant morphisms. More precisely, suppose G is a reductive linear algebraic group over an algebraically closed field k, X and Y are quasi‐projective varieties endowed with G‐actions, φ:X→Y is a G‐equivariant projective morphism, the G‐action on Y is linearized in the ample line bundle M, and the G‐action on X is linearized in the φ‐ample line bundle L. For any positive integer n, there is an induced linearization of the G‐action on X in the line bundle L⊗φ★(M⊗n). If Y is projective and n≫0, the set of points in X that are semistable with respect to this linearization is contained in the preimage under φ of the set of points in Y that are semistable with respect to the given linearization in M. The same statement is trivially also true, if Y is affine and M=OY. In this note, we show by means of an example that the statement does not hold for arbitrary quasi‐projective varieties Y. This shows that a claim by Hu of the contrary is not true. Relative geometric invariant theory plays a role in the construction and study of degenerations of moduli spaces. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the warped product pointwise semi‐slant submanifolds which are isometrically immersed into Sasakian manifolds. First, we prove some characterizations results in terms of the shape operator, under which influence a pointwise semi‐slant submanifold of a Sasakian manifold can be reduced to a warped product submanifold. Then, we determine a geometric inequality for the second fundamental form regarding to intrinsic invariant and extrinsic invariant using the Gauss equation instead of the Codazzi equation. Evenmore, we give some applications of this inequality into Sasakian space forms, and we will investigate the status of equalities in the inequality. As a particular case, we provide numerous applications of the Green lemma, the Laplacian of warped functions and some partial differential equations. Some triviality results for connected, compact warped product pointwise semi‐slant submanifolds of Sasakian space form by means of Hamiltonian and the kinetic energy of warped function involving boundary conditions are established. [ABSTRACT FROM AUTHOR]

We prove that if X:Mn→Hn×R, n⩾3, is an orientable, complete immersion with finite strong total curvature, then X is proper and M is diffeomorphic to a compact manifold M¯ minus a finite number of points q1,⋯qk. Adding some extra hypothesis, including Hr=0, where Hr is a higher order mean curvature, we obtain more information about the geometry of a neighbourhood of each puncture. The reader will also find in this paper a classification result for the hypersurfaces of Hn×R which satisfy Hr=0 and are invariant by hyperbolic translations and a maximum principle in a half‐space for these hypersurfaces. [ABSTRACT FROM AUTHOR]

We establish an explicit expression for the smallest non‐zero eigenvalue of the Laplace–Beltrami operator on every homogeneous metric on the 3‐sphere, or equivalently, on SU(2) endowed with left‐invariant metric. For the subfamily of three‐dimensional Berger spheres, we obtain a full description of their spectra. We also give several consequences of the mentioned expression. One of them improves known estimates for the smallest non‐zero eigenvalue in terms of the diameter for homogeneous 3‐spheres. Another application shows that the spectrum of the Laplace–Beltrami operator distinguishes up to isometry any left‐invariant metric on SU(2). It is also proved the non‐existence of constant scalar curvature metrics conformal and arbitrarily close to any non‐round homogeneous metric on the 3‐sphere. All of the above results are extended to left‐invariant metrics on SO(3), that is, homogeneous metrics on the three‐dimensional real projective space. [ABSTRACT FROM AUTHOR]

The dynamics of the 2D Ricker equation depend on the positioning of its isoclines. We give a complete description for the parameter values where the two isoclines overlap. In this case, we show the plane is covered by a family of invariant curves and that the restriction to each is conjugate to a family of one‐dimensional population maps. Properties of these 1D maps, including global stability, are established, and we discuss the implications for the 2D Ricker equation. [ABSTRACT FROM AUTHOR]

Let A be an abelian variety. We introduce A‐equivariant Grothendieck rings and A‐equivariant motivic Hall algebras, and endow them with natural integration maps to the ring of dual numbers. The construction allows a systematic treatment of reduced Donaldson–Thomas (DT) invariants by Hall algebra techniques. We calculate reduced DT invariants for K3×E and abelian threefolds for several imprimitive curve classes. This verifies (in special cases) multiple cover formulas conjectured by Oberdieck–Pandharipande and Bryan–Oberdieck–Pandharipande–Yin. [ABSTRACT FROM AUTHOR]

In this article, we introduce an invariant‐region‐preserving (IRP) limiter for the p‐system and the corresponding viscous p‐system, both of which share the same invariant region. Rigorous analysis is presented to show that for smooth solutions the order of approximation accuracy is not destroyed by the IRP limiter, provided the cell average stays away from the boundary of invariant region. Moreover, this limiter is explicit, and easy for computer implementation. A generic algorithm incorporating the IRP limiter is presented for high order finite volume type schemes as long as the evolved cell average of the underlying scheme stays strictly within the invariant region. For high order discontinuous Galerkin (DG) schemes to the p‐system, sufficient conditions are obtained for cell averages to stay in the invariant region. For the viscous p‐system, we design both second and third order IRP DG schemes. Numerical experiments are provided to test the proven properties of the IRP limiter and the performance of IRP DG schemes. [ABSTRACT FROM AUTHOR]

The standard method of joint probability distribution functions, so crucial for the development of direct methods, has been revisited and updated. It consists of three steps: identification of the reflections which may contribute to the estimation of a given structure invariant or seminvariant, calculation of the corresponding joint probability distribution, and derivation of the conditional distribution of the invariant or seminvariant phase given the values of some diffracted amplitudes. In this article the conditional distributions are derived directly without passing through the second step. A good feature of direct methods is that they may work in the absence of any prior information: that is also their weakness. Different types of prior information have been taken into consideration: interatomic distances, interatomic vectors, Patterson peaks, structural model. The method of directly deriving the conditional distributions has been applied to those cases. Some new formulas have been obtained estimating two‐, three‐ and four‐phase invariants. Special attention has been dedicated to the practical aspects of the new formulas, in order to simplify their possible use in direct phasing procedures. Direct methods techniques are revisited and new mathematical approaches are described. [ABSTRACT FROM AUTHOR]

Signal processing on graphs using adjacency matrix (as opposed to more traditional graph Laplacian) results in an algebraic framework for graph signals and shift invariant filters. This can be seen as an example of the algebraic signal processing theory. In this study, the authors examine the concepts of homomorphism and isomorphism between two graphs from a signal processing point of view and refer to them as GSP isomorphism and GSP homomorphism, respectively. Collectively, they refer to these concepts as structure preserving maps (SPMs). The fact that linear combination of signals and linear transforms on signals are meaningful operations has implications on the GSP isomorphism and GSP homomorphism, which diverges from the topological interpretations of the same concepts (i.e. graph isomorphism and graph homomorphism). When SPMs exist between two graphs, signals and filters can be mapped between them while preserving spectral properties. They examine conditions on adjacency matrices for such maps to exist. They also show that isospectral graphs form a special case of GSP isomorphism and that GSP isomorphism and GSP homomorphism is intrinsic to resampling and downsampling process. [ABSTRACT FROM AUTHOR]

We prove that the moduli space of polarized K3 surfaces of genus 11 with n marked points is unirational when n⩽6 and uniruled when n⩽7. As a consequence we settle a long standing but not proved assertion about the unirationality of M11,n for n⩽6. We also prove that the moduli space of polarized K3 surfaces of genus 11 with n⩾9 marked points has non‐negative Kodaira dimension. [ABSTRACT FROM AUTHOR]

A polynomial counterpart of the Seiberg–Witten invariant associated with a negative definite plumbing 3‐manifold has been proposed by earlier work of the authors. It is provided by a special decomposition of the zeta‐function defined by the combinatorics of the manifold. In this article we give an algorithm, based on multivariable Euclidean division of the zeta‐function, for the explicit calculation of the polynomial, in particular for the Seiberg–Witten invariant. [ABSTRACT FROM AUTHOR]

Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson–Schensted bijection, which is a variant of the type C Robinson–Schensted bijection between pairs of same‐shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson–Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of the exotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces. [ABSTRACT FROM AUTHOR]

We entirely classify definable sets up to definable bijections in Z‐groups, where the language is the one of ordered abelian groups. From this, we deduce, among others, a classification of definable families of bounded definable sets. [ABSTRACT FROM AUTHOR]

A general nonlinear model of illiquid markets with feedback effects is considered. This equation with 2 free functional parameters contains as partial cases the classical Black‐Scholes equation, Schönbucher‐Wilmott equation, and Sircar‐Papanicolaou equation of option pricing. We obtain here the complete group classification of the equation, and for every parameters specification we obtain the principal Lie algebra and its optimal system of 1‐dimensional subalgerbras. For every such subalgebra we calculate the invariant submodel and invariant solution, when it is possible. Thus, the series of invariant submodels and invariant solutions are derived for the considered nonlinear model. [ABSTRACT FROM AUTHOR]

In this work, we derive important properties regarding matrix invariants that occur in the theory of differential equations with reflection. [ABSTRACT FROM AUTHOR]

In this paper, a class of switching systems which have an invariant conic x2+cy2=1,c∈R, is investigated. Half attracting invariant conic x2+cy2=1,c∈R, is found in switching systems. The coexistence of small-amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems is proved. [ABSTRACT FROM AUTHOR]

Summary: To achieve high accuracy, conventional time integration methods require small time steps, especially when applied to large‐scale finite element systems. The use of small time steps calls for a large number of computations. In this paper, a strategy is proposed to construct an efficient and highly precise time integration method for linear time‐invariant systems. Adopting the multi‐substep notion, the highly precise time integration method creates an integrated amplification matrix prior to recursive calculations by multiplying the amplification matrices of N equal substeps, where a 2m algorithm and the method of storing incremental matrix are respectively employed to reduce computational cost and rounding error. On the basis of the Newmark method, two highly precise symplectic schemes and one highly precise dissipative scheme are developed. In addition, the consistent stability of the used Newmark schemes is presented. The free vibration of a three‐dimensional truss, the impact of a particle against a rod, and the forced vibrations of a rectangular thin plate and a stiff system are investigated to validate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Symbolic gestures are the hand postures with some conventionalized meanings. They are static gestures that one can perform in a very complex environment containing variations in rotation and scale without using voice. The gestures may be produced in different illumination conditions or occluding background scenarios. Any hand gesture recognition system should find enough discriminative features, such as hand-finger contextual information. However, in existing approaches, depth information of hand fingers that represents finger shapes is utilized in limited capacity to extract discriminative features of fingers. Nevertheless, if we consider finger bending information (i.e., a finger that overlaps palm), extracted from depth map, and use them as local features, static gestures varying ever so slightly can become distinguishable. Our work here corroborated this idea and we have generated depth silhouettes with variation in contrast to achieve more discriminative keypoints. This approach, in turn, improved the recognition accuracy up to 96.84%. We have applied Scale-Invariant Feature Transform (SIFT) algorithm which takes the generated depth silhouettes as input and produces robust feature descriptors as output. These features (after converting into unified dimensional feature vectors) are fed into a multiclass Support Vector Machine (SVM) classifier to measure the accuracy. We have tested our results with a standard dataset containing 10 symbolic gesture representing 10 numeric symbols (0-9). After that we have verified and compared our results among depth images, binary images, and images consisting of the hand-finger edge information generated from the same dataset. Our results show higher accuracy while applying SIFT features on depth images. Recognizing numeric symbols accurately performed through hand gestures has a huge impact on different Human-Computer Interaction (HCI) applications including augmented reality, virtual reality, and other fields. [ABSTRACT FROM AUTHOR]

Summary: Identification of structural modal parameters under seismic excitation using operational modal analysis (OMA) is a challenging task because it violates the basic assumptions of OMA: linear time‐invariant model, stationary white noise input, and adequately long data. The consequence is significant uncertainties associated with the identified modal parameters. This study aims at developing an algorithm to quantify these uncertainties from a Bayesian perspective. Representing the structure and the seismic excitation by a state‐space model, a probabilistic OMA scheme is formulated. The analytical solution for the posterior statistics is not achievable, and a Gibbs sampling algorithm is developed to provide an efficient and robust numerical solution appropriate for practical applications. The performance of the proposed method is validated by identifying a shear‐type building using simulated response data under 4 recorded earthquake motions, and a supertall building—the One Rincon Hill in San Francisco—using field‐recorded data under seismic and ambient excitations. The computed posterior distributions of modal parameters represent the knowledge extracted from the measured data; they can be reliably used for model validation and health monitoring. [ABSTRACT FROM AUTHOR]

Let S=K[x1,...,xn] denote the polynomial ring in n variables over a field K with each degxi=1 and let I⊂S be a homogeneous ideal of S with dimS/I=d. The Hilbert series of S/I is of the form hS/I(λ)/(1−λ)d, where hS/I(λ)=h0+h1λ+h2λ2+⋯+hsλs with hs≠0 is the h‐polynomial of S/I. It is known that, when S/I is Cohen–Macaulay, one has reg(S/I)=deghS/I(λ), where reg(S/I) is the (Castelnuovo–Mumford) regularity of S/I. In the present paper, given arbitrary integers r and s with r≥1 and s≥1, a monomial ideal I of S=K[x1,...,xn] with n≫0 for which reg(S/I)=r and deghS/I(λ)=s will be constructed. Furthermore, we give a class of edge ideals I⊂S of Cameron–Walker graphs with reg(S/I)=deghS/I(λ) for which S/I is not Cohen–Macaulay. [ABSTRACT FROM AUTHOR]

We prove that the bigraded, colored Khovanov–Rozansky type A link and tangle invariants are functorial with respect to link and tangle cobordisms. [ABSTRACT FROM AUTHOR]

The study examines the control algorithm of a three-dimensional attitude and position of a free-floating rigid body with three thruster forces in which the force directions are fixed with respect to the body. This study provides a theory to develop a control method of an underactuated satellite with the minimum thruster number. In the procedure, three switching controllers are used in conjunction with motion planning in the final angular-rate deceleration phase to individually control the six state variables to the target values. The switching controllers have a hierarchical structure by using invariant manifolds as switching surfaces. The state variables in higher class manifolds that include lower class ones are adjusted by repeatedly adding intentional disturbances while the lower class state variables are returned to the original values by using lower class invariant manifolds. This study describes methods to define the invariant manifolds and also the intentional disturbance for achieving the forementioned control strategy. Finally, the motion planning in the angular-rate deceleration phase from a remained single-axis rotation finalizes the six state values of the body to the target values. Numerical simulations verify the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: Shelstad's character identity is an equality between sums of characters of tempered representations in corresponding L‐packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic operators in the framework of K‐theory. Our geometric proof of the character identity is evidence that index theory can play a role in the classification of group representations via the Langlands program. [ABSTRACT FROM AUTHOR]

Abstract: We study various classes of random processes defined on the regular tree Td that are invariant under the automorphism group of Td. The most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov chains and a new class that we call typical processes. Using Glauber dynamics on processes we give a sufficient condition for a branching Markov chain to be factor of i.i.d. [ABSTRACT FROM AUTHOR]

Abstract: We investigate in detail a homomorphism which we call the 2‐Selmer signature map from the 2‐Selmer group of a number field K to a nondegenerate symmetric space, in particular proving the image is a maximal totally isotropic subspace. Applications include precise predictions on the density of fields K with given narrow class group 2‐rank and with given unit group signature rank. In addition to theoretical evidence, extensive computations for totally real cubic and quintic fields are presented that match the predictions extremely well. In an Appendix with Richard Foote, we classify the maximal totally isotropic subspaces of orthogonal direct sums of two nondegenerate symmetric spaces over perfect fields of characteristic 2 and derive some consequences, including a mass formula for such subspaces. [ABSTRACT FROM AUTHOR]

There are various models of gravitation: the metrical Hilbert‐Einstein theory, a wide class of intrinsically Lorentz‐invariant tetrad theories (generally covariant in the space‐time sense), and many gauge models based on various internal symmetry groups (Lorentz, Poincare, GL ( n , R ) , SU ( 2 , 2 ) , GL ( 4 , C ), etc). The gauge models are usually preferred but nevertheless it is an interesting idea to develop the class of GL ( 4 , R )‐invariant (or rather GL ( n , R )‐invariant) tetrad (n‐leg) generally covariant models. This is done below and motivated by our idea of bringing back to life the Thales of Miletus concept of affine symmetry. Formally, the obtained scheme is a generally covariant tetrad (n‐leg) model, but it turns out that generally covariant and intrinsically affinely invariant models must have a kind of nonaccidental Born‐Infeld‐like structure. Let us also mention that they, being based on tetrads (n‐legs), have many features common with continuous defect theories. It is interesting that they possess some group‐theoretical solutions and more general spherically symmetric solutions, discussion of which is the main new result presented in this paper, including the applications of the 't Hooft‐Polyakov monopoles in the generally covariant theories, which enables us to find some rigorous solutions of our strongly nonlinear equations. It is also interesting that within such a framework, the normal‐hyperbolic signature of the space‐time metric is not introduced by hand but appears as a kind of solution, rather integration constants, of differential equations. Let us mention that our Born‐Infeld scheme is more general than alternative tetrad models. It may be also used within more general schemes, including also the gauge ones. [ABSTRACT FROM AUTHOR]

The nonexistence of global solutions for semirelativistic equations with nongauge invariant power type nonlinearity is revisited by a relatively direct way with a pointwise estimate of fractional derivative of some test functions. [ABSTRACT FROM AUTHOR]

Approximate symmetries, which are admitted by the perturbed KdV equation, are obtained. The optimal system of one-dimensional subalgebra of symmetry algebra is obtained. The approximate invariants of the presented approximate symmetries and some new approximately invariant solutions to the equation are constructed. Moreover, the conservation laws have been constructed by using partial Lagrangian method. [ABSTRACT FROM AUTHOR]

The dynamical symmetries of 1+1-dimensional Matrix Partial Differential Equations with a Calogero potential (with/without the presence of an extra oscillatorial de Alfaro-Fubini-Furlan, DFF, term) are investigated. The first-order invariant differential operators induce several invariant algebras and superalgebras. Besides the sl(2)⊕u(1) invariance of the Calogero Conformal Mechanics, an osp2∣2 invariant superalgebra, realized by first-order and second-order differential operators, is obtained. The invariant algebras with an infinite tower of generators are given by the universal enveloping algebra of the deformed Heisenberg algebra, which is shown to be equivalent to a deformed version of the Schrödinger algebra. This vector space also gives rise to a higher-spin (gravity) superalgebra. We furthermore prove that the pure and DFF Matrix Calogero PDEs possess isomorphic dynamical symmetries, being related by a similarity transformation and a redefinition of the time variable. [ABSTRACT FROM AUTHOR]