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Gliomas are the second-most-common primary brain tumors in dogs. Surgery and radiotherapy are established treatment approaches with similar median survival time, whereas conventional chemotherapy is burdened by severe adverse effects. Spinal and leptomeningeal spread of gliomas have been described following radiotherapy treatment alone. The purpose of this study was to evaluate the outcome for four dogs with primary high-grade gliomas in the forebrain without evidence, at diagnosis, of neoplastic invasion along the spinal cord, that were treated with concomitant chemotherapy (temozolomide) and hypofractionated volumetric-modulated arc radiotherapy (VMAT-RT). Temozolomide was selected for its radiosensitive properties, and radiotherapy dose protocols of 37 Gy in 7 fractions or 42 Gy in 10 fractions were used. After an initial complete or partial response, tumors recurred across the cranial–spinal pathway. Post-mortem macroscopic examinations confirmed swollen spinal cord and hyperemic meningeal sleeve, with nodular lesions on the meningeal surface. Microscopically, infiltration of the spinal cord and meninges by neoplastic cells (with features of oligodendrogliomas) were observed. This work seems to suggest that the entire central nervous system should be investigated in diagnostic examinations of canine gliomas. Dose-escalation trials and/or spinal cord prophylaxis treatment could also be evaluated to prevent tumor progression.

Volumetric Modulated Arc Therapy (VMAT) is a modern technique, widely used in human radiotherapy, which allows a high dose to be delivered to tumor volumes and low doses to the surrounding organs at risk (OAR). Veterinary clinics takes advantage of this feature due to the small target volumes and distances between the target and the OAR. Sparing the OAR permits dose escalation, and hypofractionation regimens reduce the number of treatment sessions with a simpler manageability in the veterinary field. Multimodal volumes definition is mandatory for the small volumes involved and a positioning device precisely reproducible with a setup confirmation is needed before each session for avoiding missing the target. Additionally, the elaborate treatment plan must pursue hard constraints and objectives, and its feasibility must be evaluated with a per patient quality control. The aim of this work is to report results with regard to brain meningiomas and gliomas, trigeminal nerve tumors, brachial plexus tumors, adrenal tumors with vascular invasion and rabbit thymomas, in comparison with literature to determine if VMAT is a safe and viable alternative to surgery or chemotherapy alone, or as an adjuvant therapy in pets.

We introduce a kind of \"perturbation\" for the Li-Keiper coefficients around the Koebe function (the K function) and establish a closed system of Equations for the LiKeiper coefficients. We then check the correctness of some of the many possible solutions offered by the system, related to the discrete derivative of order n of a function. We also report a numerical finding which support our stability conjecture, i.e., that the tiny part λtiny(n) (the fluctuations around the trend) are bounded in absolute values by γ.n, where γ is the Euler-Mascheroni constant.

We develop an expression for the Li-Keiper coefficients λn in terms of k-blocks partitions, to begin with, for low values of n. The k-blocks partitions are given in terms of our cluster functions φn and the main point of this work lies in the emergence of an alternating sequence of values converging toward values of λn near the true values, i.e., increasing the index k of the blocks one obtains an increasing range of positivity of the Li-Keiper coefficients. With the contribution of k = 1 and k = 2 blocks, positivity of the λn is reached already until n = 26-27. The treatment is given here until k = 4 blocks up to n = 30. λn are all found to be positive.

In connection with the binomial transform, we establish a concrete asymptotic expression for the (trend, tiny part and complete) Li-Keiper coefficients in terms of elementary functions. The tiny fluctuations are computed up to n=5000 and a careful numerical analysis allows the derivation of an asymptotic formula using the extreme values of such fluctuations which is correct up to n=100000. Such an asymptotic expression - do not assume - but it is in agreement with the assumption of the truth of the Riemann Hypothesis.

Starting with a rigorous integral Equality, i.e., an integral over the logarithm of the modulus of the zeta function (an Equivalent of the Riemann Zeta Function having no zeros for Re(s) > 0.9) with a density (reciprocal of an hyperbolic function of parameter a, at Re(s) = 0.9), we define the Primitive Riemann Wave and compute some of its zeros (nodes). Alternatively, we study such a sequence of zeros by means of the Gauss -Lucas Theorem: attractive as well as repulsive points of the associated map are given up to t = 50, where t is the height in s = 0.9+i.t. We also report some numerical results for the case of vanishing parameter a, i.e., a→0 in the reciprocal hyperbolic measure. The numerical results suggest that the Primitive Riemann Wave has zeros extending to infinity i.e., that its convex hull contain in fact, the infinity of zeros of the Integrand, i.e., that no symmetry breaking occurs, supporting that the node of the Primitive Riemann Wave extends to infinity (with a vanishing amplitude if a > 0).

In the first part of this work (realized as a comment), we add the plots of two more values of the Li-Keiper coefficients λ5 and λ6, computed as in our recent work where the first four values were in particular given. This for the trend as well as for the oscillating path (\"tiny,\" the term coined by Maslanka in his pioneering work). Then, in the second part looking at the tiny oscillations, we propose a \"numerical conjecture\" in a more strong form, i.e., with a logarithmic behaviour and carry out a short numerical experiment on the new \"numerical conjecture.\"

In this note, we consider a constant c related to the Glaisher-Kinkelin constant, obtained by a shift of the infinite system of Equations for the Li-Keiper coefficients and related to the Euler formula at s = 2, i.e. ζ(2) = π2/6 for the Zeta function. The aim of the experiment we are carrying out is to relate the Primes with the tiny part (the fluctuations) of the Li-Keiper coefficients as well as with the trend part of them. The contribution of the series for the tiny part or for the trend part are thus related to a series for the primes and we obtain in this ways a upper and a lower bound of the constant c, which will be given with some decimals. Other relations may be obtained along the same lines and additional examples are given.

Background Magnetic resonance imaging (MRI) patterns of canine cervical hydrated nucleus pulposus extrusion (HNPE) have been described by a few reports, but the correlation between microsurgical and MRI features has never been investigated. The aim of this study was to compare the MRI features of HNPE with microsurgical findings and cytological outcomes and also to investigate the anatomical and pathophysiological aspects of the disease. Methods A prospective clinical study was conducted in 36 dogs suffering from HNPE. The diagnosis was based on high-field MRI findings of ventral extradural lesions, adjacent to the dorsal aspect of intervertebral discs, characterised by high signal intensity in T2-weighted sequences and hypointensity in T1-weighted sequences. MRI images were analysed with regard to the intervertebral space involved, the grading of spinal cord compression, the signal intensity and distribution of the material, and the thickness and signal intensity of the involved discs. All patients underwent microsurgical decompression and direct observations were recorded and films of the surgical procedure analysed. Results The majority of patients had acute onset of clinical signs (78 %), the patient did not exhibit signs of pain in 75 % of dogs and neurological deficits varied from slight tetraparesis (33 %) to tetraplegia (28 %). The localization of the extruded disc material was ventral relative to the dorsal longitudinal ligament that was lifted dorsally and appeared intact at the site of compression. Direct microsurgical observations of the HNPE sites showed that extruded disc material was collected within the fibres of the dorsal longitudinal ligament. The consistency was gelatinous in 42 %, water-like in 33 %, and lumpy liquid in 25 % of cases. Cytological samples did not detect the presence of inflammation, bacteria, fungi, neoplastic cells or foreign material. Conclusions Microsurgical features of HNPE suggest that the extruded disc is collected within the fibres of the dorsal longitudinal ligament and this may explain the typical MRI appearance of this disease. Further pathophysiological studies are needed to investigate why the cervical nucleus pulposus extrusion appears to occur without obvious trauma.

Starting with an infinite set of non linear Equations for the Li-Keiper coefficients, we first specify a lower bound emerging from the infinite set and give a characterization of it. Then, we propose a possible new upper and lower bound for the coefficients in few of the partitions occurring in the cluster functions furnishing in a nonlinear way the coefficients. A numerical experiment up to n=15 confirms the proposed bounds and an experiment, i.e. the counting of the zeros in the binary representation of an integer for a constant related to the Glaisher-Kinkelin constant is also given up to n=32.