Search

We consider regions obtained from 120 degree rotationally invariant hexagons by removing a core and three equal satellites (all equilateral triangles) so that the resulting region is both vertically symmetric and 120 degree rotationally invariant, and give simple product formulas for the number of their lozenge tilings. We describe a new method of approach for proving these formulas, and give the full details for an illustrative special case. As a byproduct, we are also able to generalize this special case in a different direction, by finding a natural counterpart of a twenty year old formula due to Ciucu, Eisenkölbl, Krattenthaler, and Zare, which went unnoticed until now. The general case of the original problem will be treated in a subsequent paper. We then work out consequences for the correlation of holes, which were the original motivation for this study. [ABSTRACT FROM AUTHOR]

The simple product formulae for derivatives of scalar functions raised to different powers are generalized for functions which take values in the set of symmetric positive definite matrices. These formulae are fundamental in derivation of various non-linear estimates, especially in the PDE theory. To get around the non-commutativity of the matrix and its derivative, we apply some well-known integral representation formulas and then we make an observation that the derivative of a matrix power is a logarithmically convex function with respect to the exponent. This is directly related to the validity of a seemingly simple inequality combining the integral averages and the inner product on matrices. The optimality of our results is illustrated on numerous examples. [ABSTRACT FROM AUTHOR]

Backdoor sets for the class CNF(2) of CNF-formulas in which every variable has at most two occurrences are studied in terms of parameterized complexity. The question whether there exists a CNF(2)-backdoor set of size k is hard for the class W[2], for both weak and strong backdoors, and in both cases it becomes fixed-parameter tractable when restricted to inputs in d-CNF for a fixed d. Besides that, it is shown that the problem of finding weak backdoor sets is W[2]-complete, for certain tractable cases. These are the first completeness results in lower levels of the W-hierarchy for any backdoor set problems. [ABSTRACT FROM AUTHOR]

The fuel derived from algae is a promising alternative that has been studied for supplying the global demand for fuels. The highly proteinaceous microalgae Chlamydomonas reinhardtii has some attributes that can make it an interesting organism for large-scale biomass generation such as: high growth rates, homogeneous biochemical characteristics, robustness and high adaptability, and it can grow both in the light and dark. Thus, in this work the Chlamydomonas reinhardtii was characterized using several techniques, to analyze its potential as feedstock in fuel and chemical production. Analytical pyrolysis was performed with and without catalysts. The tests were performed at 450 °C, 550 °C, 650 °C, and 750 °C in a Pyroprobe CDS 5200. For the catalytic tests a ratio of 1:2 (catalyst:biomass) was used. Non-catalytic pyrolysis indicated that the increase in the temperature increased the amount of hydrocarbon compounds and decreased oxygenated compounds. The bio-oil produced from catalytic pyrolysis showed a decrease in the nitrogenated compounds, which indicate that the addition of a hydrotalcite catalyst that can improve the bio-oil composition. [ABSTRACT FROM AUTHOR]

The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse's formula, by using factorial Schur functions and a generalization of the Hillman–Grassl correspondence , respectively. The main new results are two different q -analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. We establish explicit bijections between these objects and families of integer arrays with certain nonzero entries, which also proves the second formula. [ABSTRACT FROM AUTHOR]

We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang–Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of ( 2 n + 1 ) × ( 2 n + 1 ) DASASMs is ∏ i = 0 n ( 3 i ) ! ( n + i ) ! , and a conjecture of Stroganov from 2008 that the ratio between the numbers of ( 2 n + 1 ) × ( 2 n + 1 ) DASASMs with central entry −1 and 1 is n / ( n + 1 ) . Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved. [ABSTRACT FROM AUTHOR]

We consider the hypergeometric translation operators associated to the Cherednik operators and the Heckman-Opdam theory on Rd introduced by the author in [6]. Under some conditions on the root system and the multiplicity function, we prove in this paper that these operators are positivity preserving and allow positive integral representations. In particular, we deduce for the Opdam-Cherednik kernel and the Heckman-Opdam hypergeometric function the following main results:. (i) Their product formulas are positive integral transforms. (ii) We obtain for them best estimates. [ABSTRACT FROM AUTHOR]

We consider the hypergeometric translation operators associated to the Cherednik operators and the Heckman-Opdam theory on Rd introduced by the author in [6]. Under some conditions on the root system and the multiplicity function, we prove in this paper that these operators are positivity preserving and allow positive integral representations. In particular, we deduce for the Opdam-Cherednik kernel and the Heckman-Opdam hypergeometric function the following main results:. (i) Their product formulas are positive integral transforms. (ii) We obtain for them best estimates. [ABSTRACT FROM AUTHOR]

Using the combinatorial description of shuffle product, we prove or reformulate several shuffle product formulas of multiple zeta values, including a general formula of the shuffle product of two multiple zeta values, some restricted shuffle product formulas of the product of two multiple zeta values, and a restricted shuffle product formula of the product of n multiple zeta values. [ABSTRACT FROM AUTHOR]

In this paper we derive a nice general formula for the local integrals of triple product formula whenever one of the representations has sufficiently higher level than the other two. As an application we generalize Venkatesh and Woodbury's work on the subconvexity bound of triple product L-function in level aspect, allowing joint ramifications, higher ramifications, general unitary central characters, and general special values of local epsilon factors. [ABSTRACT FROM AUTHOR]

We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods( Specifically, we show that a d-sparse Hamiltonian H acting on n qubits can be simulated for time t with precision ∊ using O(τ (log(τ=∊)= log log(τ=∊))) queries and O(τ (log2(τ=∊)= log log(τ=∊))n) additional 2-qubit gates, where τ D d2‖H‖maxt( Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for timevarying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian Our algorithm is based on a significantly improved simulation of the continuous- and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error We also simplify the analysis of this conversion, avoiding the need for a complex fault-correction procedure Our simplification relies on a new form of 'oblivious amplitude amplification' that can be applied even though the reflection about the input state is unavailable Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error [ABSTRACT FROM AUTHOR]

We study descent properties of Jacob Lurie's topological chiral homology. We prove that this homology theory satisfies descent for a factorizing cover, as defined by Kevin Costello and Owen Gwilliam. We also obtain a generalization of Lurie's approach to this homology theory, which leads to a product formula for the infinity 1-category of factorization algebras, and its twisted generalization. [ABSTRACT FROM AUTHOR]

We extend the notion of parking functions to parking sequences, which include cars of different sizes, and prove a product formula for the number of such sequences. [ABSTRACT FROM AUTHOR]

In this study, effects of different stabilizers on sensory properties of ice cream were investigated by combined design which is a versatile experimental design technique because of combination of mixture design and process factor. For this aim, different product formulations were composed depending on the combined design points and sensory analyses were conducted. In the study, combined design was composed by two mixture designs and one numeric process factor. Totally, 270 different ice cream samples which have different formulations were prepared and evaluated by a 10 trained panelists in terms of taste-odor, color-appearance, texture-consistence and overall acceptance using a hedonic scale evaluation form. The panel scores were subjected to optimization using Nelder-Mead Simplex Algorithm. The best five different product formulations having different desirability values were composed. The results of this study showed that the ingredient proportion and aging time could be optimized using combined design effectively. Use of salep and guar gum among the stabilizers provided an increment on the desirability of the ice cream samples and aging time as a process factor showed a significant effect on the ice cream acceptability. Practical Applications Combined design is a quite effective mathematical approach for the optimization of the processing variables containing mixture components and process factors. So, it could be used efficiently to model the ingredients of ice cream processing. The results of this study showed that the ingredient proportion and aging time could be optimized using combined design effectively. [ABSTRACT FROM AUTHOR]

Rook theory has been investigated by many people since its introduction by Kaplansky and Riordan in 1946. Goldman, Joichi, and White in 1975 showed that the sum over k of the product of the ( n − k ) th rook numbers multiplied by the k th falling factorial polynomials factorize into a product. In the sequel, different types of generalizations and analogues of this product formula have been derived by various authors. In 2008, Miceli and Remmel constructed a rook theory model involving augmented rook boards in which they showed the validity of a general product formula which can be specialized to all other product formulas that so far have appeared in the literature on rook theory. In this work, we construct an elliptic extension of the q -analogue of Miceli and Remmel’s result. Special cases yield elliptic extensions of various known rook theory models. [ABSTRACT FROM AUTHOR]

In this article we obtain the meromorphic continuation of multiple zeta functions, together with a complete list of their poles and residues, by means of an elementary and simple translation formula for these multiple zeta functions. The use of matrices to express this translation formula leads, in particular, to a succinct description of the residues of the multiple zeta functions. We conclude our paper by introducing certain interesting weighted variants of multiple zeta functions. They are shown to behave particularly nicely with respect to product formulas and location of poles. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish a product formula for the generalizedq-Bessel function which is a generalization of the knownq-Bessel functions of kind 1,2,3, the modifiedq-Bessel functions of kind 1,2,3, and the newq-analogy of the modified Bessel function presented and studied by Mansour and Al-Shomarani. As an application of the product formula we derive Turán-type inequality for the modifiedq-Bessel function of third kind. [ABSTRACT FROM AUTHOR]

In this note, we characterize the scope of the product form solution that has been presented in the paper, `A closed form solution for the G/M/r machine interference model' (International Journal of Production Research, 1988, 26, 1811-1818). In this paper, the machine interference problem has been solved for N heterogeneous machines that are 'repaired' by a set of r operatives, which repair the machines in a service-in-random-order fashion. The authors solve the model allowing for different, independent and identically distributed times between breakdowns for each machine and different negative-exponential repair times for each machine. In this short critique, we show that the solution presented by Bunday and Khorram does not solve the classical machine interference problem unless the hypothesis of heterogeneous machines is relaxed. [ABSTRACT FROM AUTHOR]

In this paper, we give an algorithmic method of constructing inner product formulas which contain weighted inner products of holomorphic functions on the unit ball. This result covers all inner product formulas which have been proved before. As an application, we prove Zhu’s characterization theorem by an elementary method. Finally, by using one-to-one correspondence between inner product formulas and lattices with circles, we prove all inner product formulas again without any complicated computation. [ABSTRACT FROM AUTHOR]

We propose a class of models of random walks in a random environment where an exact solution can be given for a stationary distribution. The environment is cast in terms of a Jackson/Gordon-Newell network although alternative interpretations are possible. The main tool is the detailed balance equations. The difference compared to earlier works is that the position of the random walk influences the transition intensities of the network environment and vice versa, creating strong correlations. The form of the stationary distribution is closely related to the well-known product formula. [ABSTRACT FROM AUTHOR]

We investigate the connection between lozenge tilings and domino tilings by introducing a new family of regions obtained by attaching two different Aztec rectangles. We prove a simple product formula for the generating functions of the tilings of the new regions, which involves the statistics as in the Aztec diamond theorem (Elkies et al. (1992) [2,3] ). Moreover, we consider the connection between the generating function and MacMahon's q -enumeration of plane partitions fitting in a given box [ABSTRACT FROM AUTHOR]

An analytical formula for calculating the T -stress in a centrally cracked Brazilian disk under both diametric forces and confining pressure, is presented by using the weight function method. It is very accurate for α ⩽ 0.95 with 100 terms, and can be expediently used for arbitrary crack lengths and loading angles. In addition, the effect of confining pressure on the T -stress is also investigated, and it is shown that the T -stress increases with increase of confining pressure. There is a significant effect of confining pressure on the T -stress for a centrally cracked Brazilian disk with a large crack length and loading angle. [ABSTRACT FROM AUTHOR]

It has been proven that the lozenge tilings of a quartered hexagon on the triangular lattice are enumerated by a simple product formula. In this paper we give a new proof for the tiling formula by using Kuo’s graphical condensation. Our result generalizes a result of Proctor on enumeration of plane partitions contained in a “maximal staircase”. [ABSTRACT FROM AUTHOR]

We present here a set of lecture notes on quantum systems with time-dependent boundaries. In particular, we analyze the dynamics of a non-relativistic particle in a bounded domain of physical space, when the boundaries are moving or changing. In all cases, unitarity is preserved and the change of boundaries does not introduce any decoherence in the system. [ABSTRACT FROM AUTHOR]

We show that a Schur form of a real orthogonal matrix can be obtained from a full CS decomposition. Based on this fact a CS decomposition-based orthogonal eigenvalue method is developed. We also describe an algorithm for orthogonal similarity transformation of an orthogonal matrix to a condensed product form, and an algorithm for full CS decomposition. The latter uses mixed shifted and zero-shift iterations for high accuracy. Numerical examples are presented. [ABSTRACT FROM AUTHOR]

Intrigued by the product formula Πi=1n–2 Ci for the volume of the Chan-Robbins-Yuen polytope CRYn, where Ci is the ith Catalan number, we construct a family of polytopes Pm,n, indexed by m ∊ Z≥0 and n ∊ Z≥2, whose volumes are given by the product...The Chan-Robbins-Yuen polytope CRYn coincides with P0,n–2. Our construction of the polytopes Pm,n is an application of a systematic method we develop for expressing volumes of a class of flow polytopes as the number of certain triangular arrays. This method can also be used as a heuristic technique for constructing polytopes with combinatorial volumes. As an illustration of this we construct polytopes whose volumes equal the number of r-ary trees on n internal nodes, .... Using triangular arrays we also express the volumes of flow polytopes as constant terms of formal Laurent series. [ABSTRACT FROM AUTHOR]

A new algorithm is given to find product-form solutions for the joint equilibrium probabilities in a class of synchronized Markov processes. This is based on, and proved by, multiple applications of the Reversed Compound Agent Theorem (RCAT) and can describe multi-way synchronizations (seen as chains of pairwise synchronizations) that occur in a prescribed order. The length of the sequence is unbounded but finite with probability 1. Several applications are given to illustrate the methodology, which include various modes of resets in queueing networks with negative customers. In particular, it is shown that there is a type of reset that can propagate further transitions in a chain actively. Furthermore, a number of completely new product-form models, for example, where the transitions in a chain are non-homogeneous, are given. [ABSTRACT FROM AUTHOR]

Through the theory of Jack polynomials we give an iterative method for integral formula of Bessel function of typeAN−1and a partial product formula for it. [ABSTRACT FROM AUTHOR]

Abstract: A product formula of Matthews [4] for the cubic Gauss sum as defined in the Introduction will be applied to determine which of the three intervals , and contains the cubic Gauss sum , where p is a prime number congruent to one modulo 3. [Copyright &y& Elsevier]

We find new operator formulas for converting Q− P and P− Q ordering to Weyl ordering, where Q and P are the coordinate and momentum operator. In this way we reveal the essence of operators' Weyl ordering scheme, e.g., Weyl ordered operator polynomial ${_{:}^{:}}\;Q^{m}P^{n}\;{_{:}^{:}}$, where ${}_{:}^{:}$ ${}_{:}^{:}$ denotes the Weyl ordering symbol, and H is the two-variable Hermite polynomial. This helps us to know the Weyl ordering more intuitively. [ABSTRACT FROM AUTHOR]

Product formulas and inequalities involving θ functions are established. Main results of this paper were obtained using Carlson's half-argument formulas for the quotients of functions under discussion. [ABSTRACT FROM PUBLISHER]

This work investigates how consumers make preference judgments when taking into account both product form and function. In prior work, where aesthetic preference is quantified using visual conjoint methods, aesthetic preference and functional preference were handled separately. Here, we introduce a new methodology, metaconjoint analysis, for testing the hypothesis that when consumers make decisions taking into account both a product''s form and its function they employ a more complex decision-making strategy than when basing their decisions on form or function alone. We anticipate that this strategy will involve both analytical and emotional processes. When compared with participant ratings of form and function combinations across 28 subjects, the metaconjoint model is shown to have a correlation that was not statistically different from an additive model of form and function. However, unlike the additive model, the metaconjoint model gave additional information about how participants make tradeoffs between form and function. Next, we developed a novel paradigm using functional magnetic resonance imaging (fMRI) to determine what parts of the brain are primarily involved with a given tradeoff between form and function. While in the scanner, study participants were asked to make decisions in trials where only form varied, where only function varied, and where both form and function varied. Results from 14 participants suggest that choices based on products that vary in both form and function involve some unique and some common brain networks as compared to choices based on form or function alone; notably, emotion-related regions are activated during these complex decisions where form and function are in conflict. These results are consistent with the inclusion of emotion in decision-making with regards to product choice and demonstrate the feasibility of using fMRI to address questions about the mental processes underlying consumer decisions. Studying preference decisions together with their accompanying neurological activity will give engineers and designers greater insight into the consumer decision-making process. [ABSTRACT FROM AUTHOR]

The tolerance design problem involves optimizing component and assembly tolerances to minimize the total cost (sum of manufacturing cost and quality loss). Previous literature recommended using traditional response surface methodology (RSM) designs, models, and optimization techniques to solve the tolerance design problem for the worst-case scenario in which the assembly characteristic is the sum of the component characteristics. In this article, component-amount (CA) and mixture-amount (MA) experiment approaches are proposed as more appropriate for solving this class of tolerance design problems. The CA and MA approaches are typically used for product formulation problems, but can also be applied to this type of tolerance design problem. The advantages of the CA and MA approaches over the RSM approach and over the standard, worst-case tolerance-design method are explained. Reasons for choosing between the CA and MA approaches are also discussed. The CA and MA approaches (experimental design, response modeling, and optimization) are illustrated using real examples. [ABSTRACT FROM AUTHOR]

Abstract: The classical Euler decomposition theorem expressed a product of two Riemann zeta values in terms of double Euler sums. Such kind of decomposition theorem are useful to produce weighted sum formulae on double Euler sums. In this paper, we produce a more general decomposition theorem as a generalization of Euler decomposition theorem. Indeed, we obtained the shuffle product formula of the product with , and . Along with some well-known theorems from the theory of probability, we are able to build up a lot of double weighted sum formulae which express sums of multiple zeta values in terms of single zeta values. Such as where is even. [Copyright &y& Elsevier]

The use of soft rotor formula (SRF) for the level energies of K = 2 γ-band for the shape transitional even Z even N nuclei in the medium mass region is illustrated. With proper treatment, we obtained positive values of the moment of inertia and softness parameter, as opposed to negative values reported in literature. The moments of inertia of the γ-band are almost equal to the ground state band values. The systematic dependence of the softness parameter on energy ratio R is studied. The effect of the odd-even spin staggering on these parameters is studied in detail. In deformed nuclei, the same parameters for odd and even spin members yield fair energy values. [ABSTRACT FROM AUTHOR]

This paper concerns versions of the Trotter-Kato Theorem and the Chernoff Product Formula for C-semigroups in the absence of stability. Applications to ${\mathcal{A}}$-stable rational approximations of semigroups are presented. [ABSTRACT FROM AUTHOR]

Abstract: Following methods of Hayes, we state a conjectural product formula for ratios of Brumer–Stark units over real quadratic fields. [Copyright &y& Elsevier]

Abstract: We investigate properties of the determinants of tensors, and their applications in the eigenvalue theory of tensors. We show that the determinant inherits many properties of the determinant of a matrix. These properties include: solvability of polynomial systems, product formula for the determinant of a block tensor, product formula of the eigenvalues and Geršgorinʼs inequality. As a simple application, we show that if the leading coefficient tensor of a polynomial system is a triangular tensor with nonzero diagonal elements, then the system definitely has a solution in the complex space. We investigate the characteristic polynomial of a tensor through the determinant and the higher order traces. We show that the k-th order trace of a tensor is equal to the sum of the k-th powers of the eigenvalues of this tensor, and the coefficients of its characteristic polynomial are recursively generated by the higher order traces. Explicit formula for the second order trace of a tensor is given. [Copyright &y& Elsevier]

Abstract: We provide a general product formula for the solution of nonautonomous abstract delay equations. After having shown the convergence we obtain estimates on the order of convergence for differentiable history functions. Finally, the theoretical results are demonstrated by some typical numerical examples. [Copyright &y& Elsevier]

We supplement the well known upper and lower box-counting prod- uct inequalities to give the new product formula dimLB F + dimLB G ≤ dimLB (F x G) ≤ min (dimLB F + dimB G; dimB F + dimLB G) ≤ max (dimLB F + dimB G; dimB F + dimLB G) ≤ dimB (F x G) ≤ dimB F + dimB G for subsets of metric spaces. We develop a procedure for constructing sets so that the upper and lower box-counting dimensions of these sets and their product can take arbitrary values satisfying the above product formula. In particular we illustrate how badly behaved both the lower and upper box-counting dimensions can be on taking products. [ABSTRACT FROM AUTHOR]

We investigate the optimal estimation of a quantum process that can possibly consist of multiple time steps. The estimation is implemented by a quantum network that interacts with the process by sending an input and processing the output at each time step. We formulate the search for the optimal network as a semidefinite program and use duality theory to give an alternative expression for the maximum payoff achieved by estimation. Combining this formulation with a technique devised by Mittal and Szegedy we prove a general product rule for the joint estimation of independent processes, stating that the optimal joint estimation can be achieved by estimating each process independently, whenever the figure of merit is of a product form.We illustrate the result in several examples and exhibit counterexamples showing that the optimal joint network may not be the product of the optimal individual networks if the processes are not independent or if the figure of merit is not of the product form. In particular, we show that entanglement can reduce by a factor K the variance in the estimation of the sum of K independent phase shifts. [ABSTRACT FROM AUTHOR]

In his biography of Emil Artin, Richard Brauer describes the years from 1931-1941 as a time when 'Artin spoke through his students and through the members of his mathematical circle' rather than through written publications. This paper explores these seemingly quiet years when Artin immigrated to America and disseminated ideas about algebraic number theory during this time in his collaboration with George Whaples, a young American mathematician who had just completed his Ph.D. at the University of Wisconsin. The main result of their work is the use of the product formula for valuations to come up with an axiomatic characterization of both algebraic number fields and algebraic function fields with a finite field of constants. These two families of fields are exactly the fields for which class field theory is known to hold. We situate their mathematical work in the broader context of algebraic number theory and their lives within the broader historical context. [ABSTRACT FROM AUTHOR]